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%%     journal="in Recent Developments in Operator Theory and its Applications, I. Gohberg (ed.) et al., 125-163, Oper. Theory Adv. Appl. 87, Birkh\"auser, Basel, 1996",
%%     doi="10.1007/978-3-0348-9035-9_6",
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%%     }


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\begin{document}


\title{The KdV Hierarchy and Associated Trace Formulas}
\author{F.~Gesztesy}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211}
\email{gesztesyf@missouri.edu}
\author{R.~Ratnaseelan}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211}
%\email{mathgr29@mizzou1.missouri.edu}
\author{G.~Teschl}
\address{Department of Mathematics, University of Missouri,
Columbia, MO 65211}
\curraddr{Institut f\"ur Mathematik\\
Strudlhofgasse 4\\ 1090 Wien\\ Austria}
\email{Gerald.Teschl@univie.ac.at}

\keywords{KdV hierarchy, trace formulas}
\subjclass{Primary 35Q53, 34B05; Secondary 34B27, 35Q51}
\thanks{in Recent Developments in Operator Theory and its Applications, I. Gohberg (ed.) et al., 125--163, Oper. Theory Adv. Appl. {\bf 87}, Birkh\"auser, Basel, 1996}

\begin{abstract}
A natural algebraic approach to the KdV hierarchy and its
algebro-geometric finite-gap solutions is developed.  In
addition, a new
derivation of associated higher-order trace formulas in
connection with
one-dimensional Schr\"odinger operators is presented.
\end{abstract}

\maketitle


\section{Introduction}

The purpose of this paper is to advocate a most
natural algebraic
approach to hierarchies of completely integrable
evolution equations such
as the AKNS and Toda hierarchies and a systematic
treatment of associated
trace formulas.  Specifically, we shall treat in
great detail the
simplest example of these completely integrable
systems, the Korteweg-de
Vries (KdV) hierarchy, and derive the corresponding
higher-order trace
formulas for one-dimensional \schro operators.  Even
though the main
ingredients of our approach to the KdV hierarchy
(to be outlined below)
appear to be well-known, it seems to us that no
systematic attempt to
combine them all into a complete description of
the KdV hierarchy and its
algebro-geometric solutions has been undertaken in
the literature thus
far.  The principal aim of this paper is to fill
this gap and at the same
time provide the intimate connection with general
higher-order trace
formulas for the associated Lax operator.

The key ingredients just mentioned are a recursive
approach to Lax pairs
following Al'ber \cite{1}, \cite{2} (see also
\cite{9}, Ch.~12,
\cite{15}), naturally leading to the celebrated
Burchnall-Chaundy
polynomial \cite{6}, \cite{7} and hence to
hyperelliptic curves $K_g$ of
genus $g\in \bbN_0 (= \bbN \cup \{0\})$ and a
classic representation of
positive divisors of $K_g$ of degree $g$ due to
Jacobi \cite{27} and
first applied to the KdV case by Mumford
\cite{36}, Section III a).1,
with subsequent extensions due to McKean
\cite{33}.  Finally, following a
recent series of papers on trace formulas
for \schro operators
\cite{16}--\cite{19}, \cite{22}--\cite{24} we
present a new algorithm for
deriving higher-order trace formulas associated
with the KdV hierarchy.

In Section 2 we briefly review Al'ber's recursive
approach to the KdV
hierarchy.  In particular, we illustrate the role
of commuting
differential expressions of order $2g+1$,
$g\in\bbN_0$ and 2,
respectively, in connection with the
Burchnall-Chaundy polynomial,
hyperelliptic curves $K_g$ of genus $g$ branched
at infinity, and the
equations of the stationary (i.e.,
time-independent) KdV hierarchy.
Section 3 combines Al'ber's recursion formalism
with Jacobi's
representation of positive divisors of degree
$g$ of $K_g$ as
applied to the KdV case by Mumford and McKean
and provides a detailed
construction of the stationary KdV hierarchy
and its algebro-geometric
solutions.  The principal new result of
Section 3, summarized in
\eqref{3.50}--\eqref{3.65}, concern divisors
of degree $g+1$ of $K_g$
associated with Schr\" odinger-type operators
with general boundary
conditions
of the type defined in \eqref{3.44}.  In Section 4
we present a
systematic
extension of this body of ideas to the
time-dependent
 KdV hierarchy going
beyond the standard treatment in the literature.
Especially, our
$t$-dependent discussion in connection with
divisors of degree $g+1$ of
$K_g$ associated with the general eigenvalue
problem \eqref{3.44} as
presented in \eqref{4.40}--\eqref{4.54} is
without precedent.  Moreover,
our proof of the theta function representation
\eqref{4.55} of the
Baker-Akhiezer function $\psi (P, x, x_0, t, t_0)$
in Theorem~\ref{t4.6},
based on the fundamental meromorphic function
$\phi (P, x, t)$ defined in
\eqref{4.15}, is new.  In Section 5 we turn to
(higher-order) trace
formulas for \schro operators associated with
general boundary conditions
(cf.\ \eqref{5.3}), a key ingredient in the
solution of inverse spectral
problems.  Unlike Sections 3 and 4, the approach
in Section 5 applies to
general (not necessarily algebro-geometric
finite-gap)
solutions of the
KdV hierarchy.  The principal new results of
Section 5 are the
(universally valid) nonlinear differential
equation \eqref{5.18} for
$\Gam^\beta (z,x)$, $\beta \in \bbR$ (defined
in \eqref{5.4}), the
resulting recursion relation \eqref{5.21}, and,
in particular, our
method of proof of Theorem~\ref{t5.3}~(i).  In
Appendix A we provide a
brief summary on hyperelliptic curves of the
KdV-type and their theta
functions and establish our basic notation used
in Sections 3 and 4.
Finally, Appendix B provides an explicit illustration
of the Riemann-Roch
theorem in connection with hyperelliptic curves
branched at infinity
which appears to be of independent interest.\\
\indent We emphasize that the methods of this paper
are widely applicable
to
$1+1$-dimensional completely integrable systems.
The corresponding
account for the Toda and Kac-van Moerbeke hierarchy
can be found in
\cite{5}.

\section{The KdV Hierarchy, Recursion
Relations, and Hyperelliptic Curves\/}

In this section we briefly review the construction
of the KdV hierarchy
using a recursive approach advocated by
Al'ber \cite{1}, \cite{2} (see
also \cite{9}, Ch.~12, \cite{15}, \cite{20})
and outline its connection
with the Burchnall-Chaundy polynomial \cite{6},
\cite{7} and associated
hyperelliptic curves branched at infinity.

Suppose
\begin{equation}
V(., t) \in C^\infty (\bbR), \;
t\in \bbR, V(x,.) \in C^1 (\bbR), \;
x\in\bbR
\lb{2.1}
\end{equation}
and consider the differential expressions (Lax pair)
\begin{equation}
L(t) = - \dfrac{d^2}{dx^2} + V(x,t),
\lb{2.2}
\end{equation}
\begin{equation}
P_{2g+1} (t) = \sum_{j=0}^g \big[ f_j (x,t)
 \dfrac{d}{dx}
- \dfrac12
f_{j,x} (x,t) \big] L(t)^{g-j},
\quad g\in\bbN_0, \; (x,t) \in\bbR^2,
\lb{2.3}
\end{equation}
where the $\{ f_j\}_{0\leq j \leq g}$ satisfy
the recursion relation
\begin{equation}
f_0 = 1,\;
f_{j,x} = - \dfrac14 f_{j-1, xxx} + Vf_{j-1,x}
+ \dfrac12 V_x
f_{j-1},\quad 1 \leq j \leq g.
\lb{2.4}
\end{equation}
Define in addition $f_{g+1}$ by
\begin{equation}
f_{g+1,x} = - \dfrac14 f_{g,xxx} + Vf_{g,x}
+ \dfrac12 V_x f_g.
\lb{2.5}
\end{equation}
Then one computes
\begin{equation}
[P_{2g+1}, L] = 2f_{g+1,x},
\lb{2.6}
\end{equation}
where $[.,.]$ denotes the commutator.  The
Lax equation
\begin{equation}
\dfrac{d}{dt} L(t) - [P_{2g+1} (t), L(t) ] = 0,
\quad t\in\bbR
\lb{2.7}
\end{equation}
is then equivalent to
\begin{equation}
\kdv_g (V) = V_t - 2f_{g+1,x}, \quad t\in\bbR.
\lb{2.8}
\end{equation}
Varying $g\in\bbN_0$ yields the KdV hierarchy
\begin{equation}
\kdv_g (V) =0,\quad g\in\bbN_0.
\lb{2.9}
\end{equation}
Explicitly, one obtains from \eqref{2.4},
\begin{align} \no
f_0 & =1 = \ti f_0,\\ \no
f_1 & = \textstyle\frac12 V + c_1 =
c_1 \ti f_0 + \ti f_1,\\  \lb{2.10}
f_2 & = - \textstyle\frac18 V_{xx} +
\textstyle\frac38 V^2 +
 c_1\textstyle\frac12 V + c_2 = c_2 \ti
f_0 + c_1 \ti f_1 + \ti f_2,\\ \no
f_3 & = \textstyle\frac1{32} V_{xxxx} -
\textstyle\frac5{16} VV_{xx} -
 \textstyle\frac5{32} V_x^2 +
\textstyle\frac5{16} V^3
+ c_2 \textstyle\frac12 V +
c_1 [ - \textstyle\frac18 V_{xx}
+ \textstyle\frac38 V^2 ] + c_3\\ \no
& = c_3 \ti f_0 + c_2 \ti f_1 + c_1 \ti f_2 +
\ti f_3,\\ \no
&\text{etc.}
\end{align}
Hence by \eqref{2.8},
\begin{align} \no
\kdv_0 (V) & = V_t - V_x =0,\\
\kdv_1 (V) & = V_t + \textstyle\frac14 V_{xxx} -
\textstyle\frac32 VV_x - c_1
 V_x,\lb{2.11}\\ \no
\kdv_2 (V) & = V_t -
\textstyle\frac1{16} V_{xxxxx} +
\textstyle\frac58 VV_{xxx}
 + \textstyle\frac54
V_x V_{xx} - \textstyle\frac{15}8 V^2 V_x
- c_2 V_x + c_1 [ \textstyle\frac14
V_{xxx} -\textstyle\frac32 VV_x],\\ \no
& \text{etc.}
\end{align}
represent the first few equations of the KdV
hierarchy.  Here $c_\ell$
denote integration constants which naturally
arise when solving
\eqref{2.4}.  Moreover, the corresponding
homogeneous KdV equations,
obtained by taking all integration constants
equal to zero,
$c_\ell \equiv 0$,
$\ell \geq 1$ are then denoted by
\setcounter{equation}{11}
\begin{equation}
\widetilde{\kdv_g} (V) : =
\kdv_g (V) \big|_{c_\ell \equiv 0, \, 1\leq
\ell
\leq g}
\lb{2.12}
\end{equation}
and similarly we denote by $\ti P_{2g+1}
:= P_{2g+1} (c_\ell \equiv 0)$,
$\ti f_j := f_j (c_\ell \equiv 0)$, etc. the
 corresponding homogeneous
quantities.

Before we turn to a discussion of the stationary
KdV hierarchy we briefly
sketch the main steps leading to
\eqref{2.3}--\eqref{2.8}.  Let $\ker
(L(t) -z)$, $z\in\bbC$ denote the two-dimensional
nullspace of $L(t) -z$
(in the algebraic sense as opposed to the functional
analytic one). We
seek a representation of $P_{2g+1} (t)$ on
$\ker (L(t) -z)$ of the form
\begin{equation}
P_{2g+1} (t) \big|_{\ker (L(t) -z)} =
F_g (z,x,t) \dfrac{d}{dx} + G_{g-1}
(z,x,t),
\lb{2.13}
\end{equation}
where $F_g$ are polynomials in $z$ of the type
\begin{align}
F_g (z,x,t) & = \sum_{j=0}^g f_{g-j} (x,t) z^j,
\lb{2.14}\\
G_{g-1} (z,x,t) & =
\sum_{j=0}^{g-1} g_{g-j} (x,t) z^j.
\lb{2.15}
\end{align}
The Lax equation \eqref{2.7} restricted to
$\ker (L(t) -z)$ then yields
\begin{align}
\begin{split}
0 & = \{ \dot L - [ P_{2g+1}, L ]\}
\big|_{\ker (L-z)}
= \{ \dot L + (
L-z) P_{2g+1}\} \big|_{\ker (L-z)}\\
& = \big\{- [ F_{g,xx} +
2G_{g-1,x} ] \dfrac{d}{dx}
+ [V_t - F_g V_x -
2(V-z) F_{g,x} - G_{g-1,xx} ] \big\}
\big|_{\ker (L-z)}
\lb{2.16}
\end{split}
\end{align}
implying
\begin{equation}
G_{g-1}  = - F_{g,x}/2
\lb{2.17}
\end{equation}
(neglecting a trivial integration constant) and
\begin{equation}
V_t = - \dfrac12 F_{g,xxx} + 2(V-z) F_{g,x}
+ V_x F_g.
\lb{2.18}
\end{equation}
Insertion of \eqref{2.14} into \eqref{2.18}
then yields \eqref{2.8}.  We
omit further details and just record a few of
the polynomials $F_g$,
\begin{align}
\begin{split}
F_0 & =1 = \ti F_0,\\
F_1 & = c_1 +\textstyle\frac12 V + z =
c_1 \ti F_0 + \ti F_1,\\
F_2 & = c_2 + c_1\textstyle\frac12 V-
\textstyle\frac18 V_{xx} +
 \textstyle\frac38 V^2 + (c_1 +
\textstyle\frac12 V) z + z^2
= c_2 \ti F_0 + c_1 \ti F_1 + \ti F_2,\\
& \text{etc.}
\lb{2.19}
\end{split}
\end{align}
One verifies
\begin{equation}
P_{2g+1} = \sum^{g}_{m=0} c_{g-m} \ti P_{2m+1},
\quad c_0 =1.
\lb{2.20}
\end{equation}

Finally, we specialize to the stationary KdV
hierarchy characterized by
$V_t =0$ in \eqref{2.9} (respectively \eqref{2.8}),
or more precisely, by
commuting differential expressions
\begin{equation}
[P_{2g+1}, L]=0
\lb{2.21}
\end{equation}
of order $2g+1$ and $2$, respectively.
Eq.\ \eqref{2.18} then becomes
\begin{equation}
F_{g,xxx} -4 (V-z) F_{g,x} -2V_x F_g =0
\lb{2.22}
\end{equation}
and upon multiplying by $F_g$ and integrating
one infers
\begin{equation}
\dfrac12 F_{g,xx} F_g - \dfrac14 F_{g,x}^2 -
(V-z) F_g^2 = R_{2g+1}(z),
\lb{2.23}
\end{equation}
where $R_{2g+1}(z)$ is of the form
\begin{equation}
R_{2g+1} (z) = \prod_{n=0}^{2g} (z-E_n),
\quad \{ E_n\}_{0 \leq n
\leq 2g} \subset \bbC.
\lb{2.24}
\end{equation}
Because of \eqref{2.21} one computes
\begin{equation}
\big[P_{2g+1} \big|_{\ker (L-z)} \big]^2 = -
\big[ \dfrac12 F_{g,xx} F_g
- \dfrac14 F_{g,x}^2 -
(V-z) F_g^2 \big] \big|_{\ker (L-z)}
=- R_{2g+1} (z).
\lb{2.26}
\end{equation}
Since $z\in\bbC$ is arbitrary, one obtains
the Burchnall-Chaundy
polynomial \cite{6}, \cite{7} relating
$P_{2g+1}$ and $L$,
\begin{equation}
-P_{2g+1}^2 = R_{2g+1} (L) =
\prod_{n=0}^{2g} (L-E_n).
\lb{2.27}
\end{equation}
The resulting hyperelliptic curve $K_g$ of
(arithmetic) genus $g$,
obtained upon one-point compactification of
the curve
\begin{equation}
y^2 = R_{2g+1} (z) = \prod_{n=0}^{2g} (z-E_n)
\lb{2.28}
\end{equation}
(cf.\ Appendix A), will be the basic ingredient
in our algebro-geometric
treatment of the KdV hierarchy in Sections 3
and 4.

The spectral theoretic content of the
polynomials $F_g$, $G_{g-1}$ is
clearly displayed in \eqref{3.34}, \eqref{3.36},
\eqref{3.39}--\eqref{3.43}.

\section{The Stationary Formalism}
\setcounter{equation}{0}

Combining the recursion formalism of Section 2
with a polynomial approach
to represent positive divisors of degree $g$ of
a hyperelliptic curve
$K_g$ of genus $g$ originally developed by
Jacobi \cite{27} and applied
to the KdV case by Mumford \cite{36},
Section~III~a).1 and McKean
\cite{33}, we provide a detailed construction
of the stationary KdV
hierarchy and its algebro-geometric solutions.
Our considerations
\eqref{3.50}--\eqref{3.65} in connection with
the general
$\beta$-boundary conditions for Schr\" odinger-type
operators in
\eqref{3.44} are new.

As indicated at the end of Section 2, the
stationary KdV hierarchy is
intimately connected with pairs of commuting
differential expressions
$P_{2g+1}$ and $L$ of orders $2g+1$ and $2$,
respectively and
hyperelliptic curves $K_g$ obtained upon one-point
compactification of
the curve
\begin{equation}
y^2 = R_{2g+1} (z) = \prod_{n=0}^{2g} (z-E_n)
\lb{3.1}
\end{equation}
described in detail in Appendix~A (whose results
and notations we shall
freely use in the remainder of this paper).
Since we are interested in
real-valued KdV solutions we now make the
additional assumption
\begin{equation}
\{E_n\}_{0 \leq n \leq 2g} \subset \bbR,\;
E_0 < E_1 < \cdots <
E_{2g}, \quad g\in\bbN_0.
\lb{3.2}
\end{equation}
Writing
\begin{equation}
F_g(z,x) = \sum_{j=0}^g f_{g-j} (x) z^j =
\prod_{j=1}^g [z-\mu_j(x)]
\lb{3.3}
\end{equation}
and combining \eqref{2.23} and \eqref{3.3}
yields
\begin{equation}
\mu'_j (x)^2 = -4 R_{2g+1} (\mu_j (x))
\prod^g_{\substack{ k=1\\ k\neq j}} 
[\mu_j(x) - \mu_k (x) ]^{-2},
\quad 1 \leq j \leq g,\; x\in\bbR.
\lb{3.4}
\end{equation}
Integrating the nonlinear first-order system
\eqref{3.4} as a vector
field on the (complex) manifold
$K_g \times \cdots \times K_g = K_g^g$,
its solution
is well-defined as long as the $\mu$'s do not
collide.  Since we focus on
real-valued solutions $V$ of the KdV hierarchy,
we may restrict the
vector field to the submanifold
$\overset{g}{\underset{j=1}{\displaystyle\times}}
\ti
\pi^{-1}
([ E_{2j-1},
E_{2j}])$ which is isomorphic to the torus
$S^1\times \cdots \times S^1 =
T^g$. Thus
\begin{equation}
\mu'_j (x) = -2 i R_{2g+1}^{1/2} (\hat \mu_j (x))
 \prod^g_{\substack{k=1\\ k\neq j}}
[ \mu_j (x) - \mu_k (x) ]^{-1},
\quad 1\leq j \leq
g, \; x\in\bbR,
\lb{3.5}
\end{equation}
with the initial conditions
\begin{equation}
\{ \hat \mu_j (x_0) \}_{1\leq j \leq g}
\subset K_g,\; \ti \pi (\hat
\mu_j (x_0)) = \mu_j (x_0) \in [E_{2j-1}, E_{2j}],
\quad 1 \leq j \leq g
\lb{3.6}
\end{equation}
for some fixed $x_0 \in \bbR$, has the unique
solution $\{ \hat \mu_j (x)
\}_{1 \leq j \leq g} \subset K_g$ satisfying
\begin{equation}
\hat \mu_j (.) \in C^\infty (\bbR, K_g),\;
\ti \pi (\hat \mu_j(x))
\in [E_{2j-1}, E_{2j} ],
\quad 1 \leq j \leq g,\; x\in\bbR.
\lb{3.7}
\end{equation}
These facts are verified using the charts
\eqref{a.9}, \eqref{a.10}
which also shows that the solution
$\hat \mu_j (x)$ changes sheets whenever
it
hits $E_{2j-1}$ or $E_{2j}$ and its projection
$\mu_j (x) = \ti \pi (\hat
\mu_j (x))$ remains trapped in $[E_{2j-1}, E_{2j}]$
for all $x\in\bbR$.

Given \eqref{3.3}, \eqref{3.5}, and \eqref{2.17}
one obtains
\begin{align}
\begin{split}
G_{g-1} (z,x) & = - \dfrac12 F_{g,x} (z,x) =
 \dfrac12 \sum_{j=1}^g \mu'_j
(x) \prod^g_{\substack{k=1\\ k\neq j}}
 [z-\mu_k (x)]\\
& = -i \sum_{j=1}^g R_{2g+1}^{1/2} (\hat \mu_j (x))
 \prod^g_{\substack{k=1\\ k\neq j}}
\big( \dfrac{z-\mu_k (x)} {\mu_j(x) - \mu_k (x) }
\big)
\lb{3.8}
\end{split}
\end{align}
and hence
\begin{align}
\begin{split}
& R_{2g+1}^{1/2} (\hat \mu_j(x)) =
\sig_j (x) R_{2g+1} (\mu_j (x))^{1/2}
= iG_{g-1} (\mu_j(x), x),\\
& \hat \mu_j (x) = (\mu_j (x),
iG_{g-1} (\mu_j (x), x)),\quad 1\leq j
\leq g.
\lb{3.9}
\end{split}
\end{align}
Moreover, since
\begin{equation}
[R_{2g+1} (z) + G_{g-1} (z,x)^2]\big|_{z=\mu_j(x)}
=0,\quad 1\leq j \leq
g,
\lb{3.10}
\end{equation}
one infers
\begin{equation}
R_{2g+1} (z) + G_{g-1} (z,x)^2 =
F_g (z,x) H_{g+1} (z,x)
\lb{3.11}
\end{equation}
for some polynomial $H_{g+1}$ in $z$ of
degree $g+1$,
\begin{equation}
H_{g+1} (z,x) =
\prod_{\ell=0}^g [z-\nu_\ell (x)].
\lb{3.12}
\end{equation}
Eqs.~\eqref{3.9}, \eqref{3.11}, and
\eqref{3.12} suggest defining
$\{ \hat \nu_\ell (x) \}_{0 \leq \ell \leq g}
\subset K_g$ by
\begin{equation}
R_{2g+1}^{1/2} (\hat \nu_\ell (x)) =
 -i G_{g-1} (\nu_\ell (x), x),\;
\hat \nu_\ell (x) = (\nu_\ell (x),
-i G_{g-1} (\nu_\ell (x), x)),\quad
0 \leq \ell \leq g.
\lb{3.13}
\end{equation}
One verifies
\begin{equation}
\nu_0 (x) \leq E_0,\; \nu_\ell (x) \in [E_{2\ell -1},
E_{2\ell}],\quad
1\leq \ell \leq g, \; x\in\bbR.
\lb{3.14a}
\end{equation}
Next, we define the fundamental meromorphic
function $\phi(P,x)$ on
$K_g$,
\begin{align}
\begin{split}
\phi (P,x) & = \dfrac{iR_{2g+1}^{1/2} (P) -
G_{g-1} (\ti \pi (P), x)}{F_g
(\ti \pi (P),x)} = \dfrac{iR_{2g+1}^{1/2} (P) +
\textstyle\frac12 F_{g,x} (\ti
 \pi
(P), x)}{F_g (\ti \pi (P), x)}\\
& = \dfrac{-H_{g+1} (\ti \pi (P), x)}
{iR_{2g+1}^{1/2} (P) + G_{g-1} (\ti
\pi (P), x)},\;
P  = (\ti \pi (P),R_{2g+1}^{1/2}(P)), \; x\in\bbR,
\lb{3.14}
\end{split}
\end{align}
with divisor $(\phi (., x))$ given by
\begin{equation}
(\phi (., x))
=\calD_{\hat \nu_0(x) \hat{\underline \nu} (x)} -
\calD_{P_\infty \hat{\underline \mu} (x)}.
\lb{3.15}
\end{equation}
Here we abbreviated
\begin{equation}
\hat{\underline \nu} (x) = ( \hat \nu_1 (x), \ldots,
\hat \nu_g (x)),\;
\hat{\underline \mu}(x) =
(\hat \mu_1 (x), \ldots, \hat \mu_g (x)).
\lb{3.16}
\end{equation}
Given $\phi (P, x)$ we define the stationary
Baker-Akhiezer (BA) function
$\psi (P, x, x_0)$, meromorphic on
$K_g\bs \{ P_\infty \}$, by
\begin{equation}
\psi(P,x,x_0) =
\exp \big[ \int_{x_0}^x \, dy \phi (P,y)\big], \quad
(x,x_0) \in\bbR^2.
\lb{3.17}
\end{equation}
Properties of $V(x)$, $\phi (P,x)$, and
$\psi (P, x, x_0)$ are summarized
in the following

\begin{lem} \lb{l3.1}
Let $P= (z,\sig R_{2g+1} (z)^{1/2})
= (\ti \pi (P),R_{2g+1}^{1/2}(P)) \in K_g \bs
\{P_\infty\}$,
$(z,x,x_0) \in \bbC\times \bbR^2$. Then
\begin{alignat}{2}
& \mbox{(i).} \qquad && V(x) = E_0 +
\sum_{j=1}^g [E_{2j-1}
+ E_{2j} -2\mu_j (x) ]. \hspace*{7cm}
\lb{3.18}\\
& \mbox{(ii).} && \phi (P,x) \mbox{satisfies
the Riccati-type
equation}\notag\\
&&& \phi_x (P,x) + \phi(P,x)^2 = V(x) -z.
\lb{3.19}\\
&\mbox{(iii).} && \psi (P,x,x_0) \mbox{ satisfies
the \schro equation
}\notag \\
&&& - \psi_{xx} (P,x,x_0) +
[V(x) -z] \psi (P,x,x_0) =0.
\lb{3.20}\\
&\mbox{(iv).} && \phi(P,x) \phi (P^*, x)
= H_{g+1} (z,x) / F_g (z,x).
\lb{3.21}\\
&\mbox{(v).} && \phi (P,x) + \phi (P^*, x)
= -2G_{g-1} (z,x) / F_g (z,x)
= F_{g,x} (z,x) / F_g (z,x).
\lb{3.22}\\
&\mbox{(vi).} && \phi (P,x) - \phi (P^*, x)
 = 2i R_{2g+1}^{1/2} (P) /
F_g (z,x).
\lb{3.23}\\
&\mbox{(vii).} && \psi (P,x,x_0) \psi (P^*, x, x_0)
= F_g (z,x) / F_g
(z,x_0).
\lb{3.24}\\
&\mbox{(viii).}
&& \psi_x (P,x,x_0) \psi_x (P^*, x, x_0)
= H_{g+1} (z,x)
/ F_g (z,x_0).\\
&\mbox{(ix).} && \psi (P,x,x_0) =
[F_g (z,x) / F_g (z,x_0) ]^{1/2} \exp
\big[iR_{2g+1}^{1/2} (P) \int_{x_0}^x \, dy
 F_g (z,y)^{-1} \big].
\lb{3.26}
\end{alignat}
\end{lem}

\begin{proof}
(i). Insert \eqref{3.3} into \eqref{2.23} and
compare the coefficient of
$z^{2g}$.
(ii).  Combine \eqref{2.17}, \eqref{2.23},
and \eqref{3.14}.
(iii).  Follows from $\psi_{xx} / \psi = \phi_x
+ \phi^2 = V-z$.
(iv). Multiply the first and third expression in
\eqref{3.14} replacing
$P$ by $P^*$ in one of the two factors.
(v), (vi) are clear from \eqref{3.14}.
(vii). Combine \eqref{3.17} and \eqref{3.22}.
(viii).  Use \eqref{3.21}, \eqref{3.24}, and
$\psi_x = \phi \psi$.
(ix). Invoke \eqref{2.17}, \eqref{3.14}, and
\eqref{3.17}.
\end{proof}

Eq.~\eqref{3.18} represents a trace formula
for the finite-gap potential
$V(x)$.  The method of proof of
Lemma~\ref{l3.1}~(i) indicates that
higher-order trace formulas associated with
the KdV hierarchy can be
obtained from \eqref{3.3} and \eqref{2.23}
comparing powers of $z$. Since
we shall derive trace formulas for general
potentials in Section 5, we
postpone the special case of finite-gap
potentials at this point and
refer to Example~\ref{e5.5}.

We also record

\begin{lem} \lb{l3.2}
Let $(z,x)\in \bbC\times \bbR$.  Then
\begin{alignat}{2}
 & \mbox{(i).}\qquad && H_{g+1} (z,x) =
 \dfrac12 F_{g,xx}
(z,x) - [V(x) -z] F_g (z,x). \hspace*{5.5cm}
\lb{3.27}\\
& \mbox{(ii).}  && H_{g+1, x}(z,x) =
-2 [V(x) -z] G_{g-1} (z,x).
\lb{3.28}
\end{alignat}
\end{lem}

\begin{proof}
(i). By \eqref{2.17}, \eqref{2.23}, and
\eqref{3.11},
\begin{align*}
& -\textstyle\frac12 F_{g,xx} =
-\textstyle\frac14 F_g^{-1} F_{g,x}^2 - (V-z) F_g
 - F_g^{-1}
R_{2g+1}\\
& = -(V-z) F_g - F_g^{-1} (R_{2g+1} + G_{g-1}^2 )
= -(V-z) F_g - H_{g+1}.
\end{align*}
(ii). By \eqref{2.17}, \eqref{2.22}, and
\eqref{3.27},
\begin{align*}
& H_{g+1, x} = -G_{g-1, xx}
- (V-z) F_{g,x} - V_x F_g\\
& = \textstyle\frac12 F_{g,xxx} -
(V-z) F_{g,x} - V_x F_g = (V-z) F_{g,x} = -2
(V-z) G_{g-1}. \qed
\end{align*}
\renewcommand{\qed}{}
\end{proof}

Explicitly, one computes from \eqref{2.4},
\eqref{2.14}, and
\eqref{3.27},
\begin{align}
\begin{split}
H_1 & = \ti H_1 = -V+z,\\
H_2 & = -c_1 V + \textstyle\frac14 V_{xx} -
\textstyle\frac12 V^2 + \big( c_1 -
 \textstyle\frac12
V\big)z + z^2 = c_1 \ti H_1 + \ti H_2,\\
H_3 & = -c_2 V + c_1 (\textstyle\frac14 V_{xx} -
\textstyle\frac12 V^2\big) -
\textstyle\frac1{16} V_{xxxx} +
\textstyle\frac38 V_x^2
+ \textstyle\frac12 VV_{xx} -
\textstyle\frac38 V^3\\
& + [c_2 - c_1 \textstyle\frac12 V +
\textstyle\frac18 V_{xx} -
\textstyle\frac18 V^2 ]z
+ [c_1 - \textstyle\frac12 V\big] z^2
+ z^3 = c_2 \ti H_1 + c_1 \ti
H_2 + \ti H_3,\\
& \text{etc.}
\lb{3.29}
\end{split}
\end{align}

We also mention the following well-known
result connecting
Dirichlet and Neumann eigenvalues.

\begin{lem}\lb{l3.3}
\cite{33} Suppose $\mu_j (x_0) \in \{ E_{2j-1},
E_{2j}\}$, $1 \leq j \leq
g$.  Then $\nu_0 (x_0) =
E_0$, $\nu_j (x_0) \in \{ E_{2j-1}, E_{2j}\} \bs
\{ \mu_j (x_0) \}$, $1 \leq j \leq g$.
Conversely, suppose $\nu_j (x_0)
\in \{ E_{2j-1}, E_{2j} \}$, $ 1\leq j \leq g$.
Then $\nu_0 (x_0) =
E_0$, $\mu_j (x_0) \in
\{ E_{2j-1}, E_{2j} \} \bs \{ \nu_j (x_0) \}$,
$1\leq j \leq g$.
\end{lem}

\begin{proof}
If $\mu_j (x_0) \in \{E_{2j-1}, E_{2j} \}$,
$1\leq j \leq g$ then
$G_{g-1} (z,x_0) =0$ in \eqref{3.11} yields
$R_{2g+1}(z) = F_g(z, x_0)
H_{g+1} (z,x_0)$ and hence proves the first
claim.  Conversely, assuming
$\nu_j (x_0) \in \{ E_{2j-1}, E_{2j}\}$,
$1\leq j \leq g$ one infers from
\eqref{3.13} that $G_{g-1} (\nu_j (x_0), x_0)
= iR_{2g+1}^{1/2} (\hat
\nu_j (x_0)) =0$, $1\leq j \leq g$, i.e.,
again $G_{g-1} (z,x_0) =0$.
Hence $R_{2g+1} (z) =
F_g (z,x_0) H_{g+1} (z,x_0)$
also proves the second
claim.
\end{proof}

Given the bounded potential $V(x)$
in \eqref{3.18}, consider the
differential
expression $\tau = - \textstyle\frac{d^2}{dx^2}+
V(x)$ and define the
 corresponding
self-adjoint \schro operator $H$ in
$L^2 (\bbR)$ by
\begin{equation}
 Hf = \tau f, \; \tau = - \dfrac{d^2}{dx^2}
+ V(x), \quad x\in\bbR,\;
 f\in\calD (H) = H^{2,2} (\bbR),
\lb{3.30}
\end{equation}
with $H^{m,n} (.)$ the usual Sobolev spaces.
The resolvent of $H$ reads
\begin{equation}
((H-z)^{-1}f)(x) = \int_{\bbR} \, dx' G (z,x,x') f(x'),
\; z\in\bbC \bs
\sig (H),\; f\in L^2 (\bbR),
\lb{3.31}
\end{equation}
where the Green's function $G (z,x,x')$ is
explicitly given by
\begin{equation}
G(z,x,x') = W(\psi_+ (z,.,x_0),
\psi_- (z,.,x_0))^{-1} \begin{cases}
\psi_+ (z,x,x_0) \psi_- (z,x',x_0),
& x \geq x'\\
\psi_+ (z,x',x_0) \psi_- (z,x,x_0),
& x\leq x'
\end{cases},
\lb{3.32}
\end{equation}
with $W(f,g) = fg' - f'g$ the Wronskian of
$f$ and $g$ and $\psi_\pm
(z,x,x_0)$ the branches of $\psi (P,x,x_0)$
in the charts $(\Pi_\pm, \ti
\pi)$.  One computes
\begin{equation}
W(\psi_+ (z,.,x_0), \psi_- (z,.,x_0)) =
(2/i) R_{2g+1} (z)^{1/2} F_g
(z,x_0)^{-1}
\lb{3.33}
\end{equation}
and
\begin{equation}
G(z,x,x) = \dfrac{i \prod_{j=1}^g [ z-\mu_j (x)]}
{2R_{2g+1} (z)^{1/2}} =
\dfrac{iF_g(z,x)}{2R_{2g+1} (z)^{1/2}} \; ,
\lb{3.34}
\end{equation}
taking into account our convention \eqref{a.5}
for $R_{2g+1} (z)^{1/2}$.
In particular, the spectrum $\sig (H)$ of $H$
is given by
\begin{equation}
\sig (H) = \bigcup_{j=0}^{g-1} [E_{2j},
E_{2j+1}] \cup [E_{2g}, \infty).
\lb{3.35}
\end{equation}
Eq.~\eqref{3.34} illustrates the spectral
theoretic content of the
polynomial $F_g (z,x)$.  Moreover, the Weyl
$m$-functions $m_\pm (z,x_0)$,
associated with the restriction of $\tau$ to
$(x_0, \pm \infty)$ with a
Dirichlet boundary condition at $x_0$, read
\begin{equation}
m_\pm (z,x_0) = \phi_\pm (z,x_0) = [\pm
iR_{2g+1} (z)^{1/2} -
G_{g-1} (z,x_0) ] F_g (z,x_0)^{-1},
\lb{3.36}
\end{equation}
where $\phi_\pm (z,x)$ denote the branches
of $\phi (P,x)$ in the charts
$(\Pi_\pm, \ti\pi)$.  As a consequence, the
Weyl $m$-matrix $M(z,x_0)$
associated with $H$ is given by (see, e.g.,
\cite{32}, Ch.~8)
\begin{align}
\begin{split}
M(z,x_0) & = \scriptstyle [m_- (z,x_0)
- m_+ (z,x_0) ]^{-1} \begin{pmatrix}
\scriptstyle m_- (z,x_0) m_+ (z,x_0)
& \scriptstyle [m_- (z,x_0) + m_+
(z,x_0)]/2\\{} \scriptstyle [m_- (z,x_0) +
m_+ (z,x_0)]/2 & \scriptstyle 1
\end{pmatrix}\\
& = \begin{pmatrix}
\pa_1 \pa_2 G(z,x_0, x_0)
& \textstyle\frac12 (\pa_1
+ \pa_2) G(z,x_0, x_0) \\
\textstyle\frac12 (\pa_1 + \pa_2 ) G (z,x_0, x_0)
& G(z,x_0, x_0)
\end{pmatrix}\\
& = \dfrac{i}{2R_{2g+1} (z)^{1/2}}
\begin{pmatrix}
H_{g+1} (z,x_0) & -G_{g-1}(z,x_0)\\
-G_{g-1} (z,x_0) & F_g (z,x_0)
\end{pmatrix},
\lb{3.37}
\end{split}
\end{align}
where
\begin{align}
\begin{split}
\pa_1 G(z,x_0, x') & =
\pa_x G(z,x,x')\big|_{x=x_0},
\; \pa_2 G(z,x,x_0)
= \pa_{x'} G (z,x,x')\big|_{x' = x_0},\\
\pa_1 \pa_2 G(z,x_0, x_0) & =
\pa_x \pa_{x'} G(z,x,x')\big|_{x=x_0=x'},\;
\mbox{ etc.}
\lb{3.38}
\end{split}
\end{align}
The corresponding self-adjoint spectral
 matrix $\rho (\lam, x_0)$, defined
by
\begin{align}
& M_{p,q} (z,x_0) =
\int_\bbR (z-\lam)^{-1} d\rho_{p,q} (\lam, x_0),
\lb{3.39}\\
\begin{split}
& \rho_{p,q} (\lam, x_0) - \rho_{p,q} (\mu,x_0)
= \lim\limits_{\del
\downarrow 0}
\lim\limits_{\eps \downarrow 0} \pi^{-1}
\int_{\mu+\del}^{\lam + \del} \,
d\nu \iim [M_{p,q} (\nu+i\eps, x_0)
],\\
& \hspace*{3in} \lam, \mu \in\bbR,
\; 1 \leq p,q \leq 2,
\lb{3.40}
\end{split}
\end{align}
explicitly reads (cf., e.g.,
\cite{32}, Ch.~8)
\begin{align}
\dfrac{d\rho_{1,1} (\lam, x_0)}{d\lam}
& = \begin{cases}
\dfrac{H_{g+1} (\lam, x_0)}
{2\pi R_{2g+1} (\lam)^{1/2}}, & \lam \in \sig
(H)^o \\
0, & \lam \in \bbR \bs \sig (H)
\end{cases},
\lb{3.41}\\
\dfrac{d\rho_{1,2} (\lam, x_0)}{d\lam} &
= \dfrac{d\rho_{2,1}(\lam,
x_0)}{d\lam} = \begin{cases}
\dfrac{-G_{g-1} (\lam, x_0)}{2\pi R_{2g+1}
(\lam)^{1/2}}, & \lam \in \sig (H)^o\\
0, & \lam \in \bbR \bs \sig (H)
\end{cases},
\lb{3.42}\\
\dfrac{d\rho_{2,2} (\lam, x_0)}{d\lam}
& = \begin{cases}
\dfrac{F_g (\lam, x_0)}{2\pi R_{2g+1} (\lam)^{1/2}},
& \lam \in \sig
(H)^o\\ 0, & \lam\in\bbR \bs\sig (H)
\end{cases}.
\lb{3.43}
\end{align}
(Here $A^o$ denotes the interior of
$A\subset \bbR$.)

Closely associated with $H$ is
$H_{x_0}^\beta$ in $L^2(\bbR)$ defined
by
\begin{align}
\begin{split}
H_{x_0}^\beta f & = \tau f,
\beta \in \bbR \cup \{ \infty \}, \quad x_0
\in\bbR,\\
f\in\calD (H_{x_0}^\beta) & =
\{ g\in L^2 (\bbR) | g, g' \in AC ([ x_0,
\pm R]) \mbox{ for all } R> 0,\\
& \quad \lim\limits_{\eps \downarrow 0
} [g' (x_0 \pm \eps) + \beta g
(x_0 \pm \eps)] =0, \; \tau g \in L^2(\bbR)\},
\lb{3.44}
\end{split}
\end{align}
with $AC_{(\loc)} (I)$ the set of
(locally) absolutely continuous
functions on $I$.  Here, in obvious notation,
$\beta = \infty$ denotes
the Dirichlet \schro operator $H_{x_0}^D =
H_{x_0}^\infty$ and $\beta =0$
the corresponding Neumann \schro operator
$H_{x_0}^N = H_{x_0}^0$.
Moreover, $H_{x_0}^\beta$ decomposes into
the direct sum of half-line
operators
\begin{equation}
H_{x_0}^\beta = H_{-, x_0}^\beta
\oplus H_{+, x_0}^\beta,\;
L^2(\bbR) = L^2 ((-\infty, x_0])
\oplus L^2 ([x_0, \infty)).
\lb{3.45}
\end{equation}
The resolvent of $H_{x_0}^\beta$ reads
\begin{equation}
((H_{x_0}^\beta -z)^{-1} f)(x) =
 \int_\bbR \, dx' G_{x_0}^\beta (z,x,x')
f(x'), \quad z\in \bbC \bs \sig (H_{x_0}^\beta),
\; f\in L^2 (\bbR),
\lb{3.47}
\end{equation}
\vspace*{-5mm}
{\allw
\begin{align}
\begin{split}
G_{x_0}^\beta (z,x,x') & = G (z,x,x') -
\dfrac{(\beta + \pa_2 )
G(z,x,x_0) (\beta + \pa_1)G(z,x_0, x')}
{(\beta + \pa_1)(\beta + \pa_2) G
(z,x_0, x_0)},\\
& \quad\hspace{3.5cm} \beta \in \bbR,
\; z\in\bbC \bs \{ \sig
(H_{x_0}^\beta)
\cup
\sig
(H)\},
\lb{3.48}
\end{split}\\
\begin{split}
G_{x_0}^\infty (z,x,x') & = G(z,x,x') -
G(z,x,x_0) G(z,x_0, x')
G(z,x_0, x_0)^{-1},\\
& \quad\hspace{4cm} z\in\bbC \bs
\{ \sig (H_{x_0}^\infty ) \cup \sig (H)
\}.
\lb{3.49}
\end{split}
\end{align}
}
Next we define the polynomial
$K_{g+1}^\beta (z,x)$, $\beta \in\bbR$ of
degree $g+1$ in $z$,
\begin{equation}
K_{g+1}^\beta (z,x) = H_{g+1} (z,x) -
2\beta G_{g-1} (z,x) + \beta^2
F_g (z,x)
= \prod_{\ell =0}^g [z-\lam_\ell^\beta (x) ],
\quad \beta \in\bbR.
\lb{3.50}
\end{equation}
In particular,
\begin{equation}
H_{g+1} (z,x) = K_{g+1}^0 (z,x), \; \nu_\ell (x)
= \lam_\ell^0 (x), \quad
0 \leq \ell \leq g.
\lb{3.51}
\end{equation}
Explicitly, one computes
{\allw
\begin{align}
\begin{split}
K_1^\beta & = \beta^2 - V + z
= \ti K_1^\beta ,\\
K_2^\beta & = c_1 (\beta^2 - V) +
\textstyle\frac14 V_{xx} - \textstyle\frac12
 V^2
+ \textstyle\frac12 \beta V_x +
\textstyle\frac12 \beta^2 V\\
& \quad + \big(c_1 + \beta^2 -
\textstyle\frac12 V \big) z + z^2 = c_1 \ti
K_1^\beta + \ti K_2^\beta,\\
K_3^\beta & = c_2 (\beta^2 - V) +
c_1 \big(\textstyle\frac12 \beta^2 V +
 \textstyle\frac12
\beta V_x + \textstyle\frac14 V_{xx} -
\textstyle\frac12 V^2\big)
-\textstyle\frac18 \beta V_{xxx}\\
&\quad + \textstyle\frac34 \beta VV_x -
\textstyle\frac18 \beta^2
V_{xx} + \textstyle\frac38 \beta^2 V^2 -
\textstyle\frac1{16} V_{xxxx}
+ \textstyle\frac38 V_x^2 +
\textstyle\frac12 VV_{xx} - \textstyle\frac38
V^3\\ & \quad + [ \textstyle\frac12 \beta V_x +
\beta^2 c_1 +
\textstyle\frac12 \beta^2 V -
\textstyle\frac12 c_1 V + c_2 + \textstyle\frac18
V_{xx} - \textstyle\frac18  V^2 ] z +
[ c_1 + \beta^2 - \textstyle\frac12 V ] z^2 + z^3\\
& = c_2\ti K_1^\beta + c_1 \ti K_2^\beta +
\ti K_3^\beta,\\
& \text{etc.}
\lb{3.52}
\end{split}
\end{align}}
\vspace*{-5mm}
Then \eqref{3.34} and
\begin{equation}
(\beta + \pa_1) ( \beta + \pa_2) G(z,x,x)
= \dfrac{iK_{g+1}^\beta
(z,x)}{2R_{2g+1} (z)^{1/2}}, \quad \beta \in\bbR
\lb{3.53}
\end{equation}
together with \eqref{3.47} and \eqref{3.48} yield
\begin{align}
\sig (H_{x_0}^\beta ) & =
 \sig (H) \cup \{ \lam_\ell^\beta (x_0)
\}_{0\leq \ell \leq g}, \quad \beta \in \bbR,
\lb{3.54}\\
\sig (H_{x_0}^\infty) &
= \sig (H) \cup \{ \mu_j(x_0) \}_{1 \leq j \leq
g},\quad \mu_j (x_0) =
\lam_j^\infty (x_0), \; 1 \leq j \leq g,
\lb{3.55}
\end{align}
with
\begin{equation}
\lam_0^\beta (x_0) \leq E_0, \;
\beta \in \bbR,\; \lam_\ell^\beta (x_0)
\in [E_{2\ell -1}, E_{2\ell}],
\; 1\leq \ell \leq g,
\quad \beta \in\bbR \cup \{ \infty\}.
\lb{3.56}
\end{equation}
Next, one verifies
\begin{align}
\begin{split}
\phi (P, x) + \beta & =
\dfrac{iR_{2g+1}^{1/2} (P) - G_{g-1} (\ti \pi
(P), x) + \beta F_g (\ti \pi (P), x)}
{F_g (\ti \pi (P), x)}\\
& = \dfrac{-K_{g+1}^\beta (\ti \pi (P), x)}
{iR_{2g+1}^{1/2} (P) + G_{g-1}
(\ti \pi (P), x) - \beta F_g (\ti \pi (P), x)},
\lb{3.57}
\end{split}
\end{align}
\begin{align}
& R_{2g+1} (z) + [G_{g-1} (z,x) -
\beta F_g (z,x)]^2 = F_g (z,x)
K_{g+1}^\beta (z,x),
\lb{3.58}\\
& [\phi (P,x) + \beta] [ \phi (P^*, x) +
\beta ] = K_{g+1}^\beta (z,x) /
F_g (z,x),
\lb{3.59}\\
\begin{split}
& [\psi_x (P,x,x_0) + \beta \psi (P, x,x_0)]
[\psi_x (P^*, x, x_0) + \beta
\psi (P^*, x, x_0)]
= K_{g+1}^\beta (z,x) / F_g (z,x_0).
\lb{3.60}
\end{split}
\end{align}
The divisor of $\phi (.,x) + \beta$,
$\beta \in\bbR$ is given by
\begin{equation}
(\phi
(.,x)+\beta) =
\calD_{\hat\lam_0^\beta (x) \hulam^\beta
(x) } - \calD_{P_\infty \humu (x)},
\lb{3.61}
\end{equation}
with
\begin{align}
\begin{split}
& R_{2g+1}^{1/2} (\hat \lam_\ell^\beta (x))
= -i G_{g-1} (\lam_\ell^\beta
(x), x) + i\beta F_g (\lam_\ell^\beta (x), x),\\
& \hat \lam_\ell^\beta (x) =
 (\lam_\ell^\beta (x), -iG_{g-1}
(\lam_\ell^\beta (x), x) +
i\beta F_g (\lam_\ell^\beta (x), x)),\:
0 \leq \ell \leq g, \; \beta\in\bbR.
\lb{3.62}
\end{split}
\end{align}
The first-order system of differential
equations for $\lam_\ell^\beta
(x)$, $\beta \in\bbR$, i.e., the analog
of \eqref{3.5} in the case
$\beta=\infty$, will be derived in the
next section (see \eqref{4.49} for
$r=0$).  Here we only record the final
result for completeness,
\begin{align}
\begin{split}
\lb{3.63}
\lam_\ell^{\beta}{}'(x) =-2i [\beta^2-V(x) +
 \lam_\ell^\beta (x)]
R_{2g+1}^{1/2} (\hat \lam_\ell^\beta (x))
 \prod^g_{\substack{m=0\\ m\neq \ell}}
 [\lam_\ell^\beta (x) - \lam_m^\beta (x)]^{-1},
\\
\ti\pi (\hat \lam_0^\beta (x)) =
 \lam_0^\beta (x) \leq E_0,\; \ti \pi
(\hat\lam_\ell^\beta (x)) =
 \lam_\ell^\beta (x) \in
[E_{2\ell-1},E_{2\ell}],\;
1 \leq \ell \leq g, \; (\beta, x)\in\bbR^2.
\end{split}
\end{align}
In particular, taking $\beta=0$ in
\eqref{3.63} then yields the
first-order system of differential
equations for $\nu_\ell (x)$, $0 \leq
\ell \leq g$. (We remark that $V(x)$ in
\eqref{3.63} has been used for reasons of
brevity only. In order to obtain a system
of differential equations for $\lam_\ell^\beta (x)$
one needs to replace $V(x)$ by the corresponding
trace formula (see, e.g., \eqref{5.23}, \eqref{5.30},
and \eqref{5.34}).)

We emphasize that due to our convention
\eqref{a.5} for $R_{2g+1}^{1/2}
(P)$, the differential equations \eqref{3.5}
and \eqref{3.63} exhibit the
well-known monotonicity properties of
$\mu_j (x)$ and $\lam_j^\beta
(x)$, $\beta \in \bbR$, $j \geq 1$ with respect to
$x\in\bbR$. For instance,
Dirichlet eigenvalues corresponding to
the right (left) half axis
$(x,\infty) \; ((-\infty, x))$ associated
with the decomposition
\eqref{3.45} are always increasing
(decreasing) with
respect to $x\in\bbR$, etc.

We conclude with the $\theta$-function
representation for $\phi (P,x)$,
$\psi (P,x,x_0)$, $V(x)$ to be derived
in Section 4 (cf.\
Theorem~\ref{t4.6}) in the general
$t$-dependent case.
\begin{align}
\begin{split}
\phi (P,x) = &-\beta +
\dfrac{\theta (\underline{\Xi}_{P_0} - \ua_{P_0}
(P_\infty) + \ual_{P_0} (\humu (x)))}
{\theta(\uxi_{P_0} -\ua_{P_0}
(P_\infty) +
\ual_{P_0} (\hulam^\beta (x)))} \bullet\\
& \bullet \dfrac{\theta (\uxi_{P_0} -
\ua_{P_0} (P) + \ual_{P_0}
(\hulam^\beta (x)))}{\theta (\uxi_{P_0}
-\ua_{P_0} (P) + \ual_{P_0}
(\humu (x)))} \exp \big[ -\int_{P_0}^P
\ome_{P_\infty,\hat \lam_0^\beta (x)}^{(3)}\big],
\lb{3.65}
\end{split}
\end{align}
\begin{align}
\begin{split}
\psi(P,x,x_0) = & \dfrac{\theta (\uxi_{P_0}
- \ua_{P_0} (P) + \ual_{P_0}
(\humu (x)))}{ \theta (\uxi_{P_0} -
\ua_{P_0} (P_\infty) + \ual_{P_0} (
\humu (x)))} \dfrac{\theta (\uxi_{P_0} -
\ua_{P_0} (P_\infty) + \ual_{P_0}
(\humu (x_0)))}{\theta (\uxi_{P_0} -
\ua_{P_0} (P) + \ual_{P_0} (\humu
(x_0)))} \bullet\\
& \bullet \exp \big[-i(x-x_0) \int_{P_0}^P
\ome_{P_\infty,0}^{(2)}\big],\quad P_0
= (E_0, 0),\; \beta \in \bbR,
\lb{3.67}
\end{split}
\end{align}
with the linearizing property of the
Abel map,
\begin{align}
& \ual_{P_0} (\humu(x)) =
\ual_{P_0} (\humu (x_0)) +
\dfrac{\uu_0^{(2)}}{2\pi} (x-x_0),
\quad (x,x_0) \in\bbR^2,
\lb{3.68}\\
& \uu_0^{(2)} = (U_{0,1}^{(2)},
\ldots, U_{0,g}^{(2)} ),\; U_{0,j}^{(2)}
= \int_{b_j} \ome_{P_\infty,0}^{(2)}, \;
1 \leq j \leq g.
\lb{3.69}
\end{align}
The Its-Matveev formula \cite{26}, \cite{4},
Ch.~3, \cite{38}, Ch.~II for
$V(x)$ then reads
\begin{align}
\begin{split}
& V(x) = E_0 + \sum_{j=1}^g (E_{2j-1} +
E_{2j} - 2\lam_j)
 -2\pa_x^2 \ln [\theta (\uxi_{P_0} -
\ua_{P_0} (P_\infty) +
\ual_{P_0} (\humu (x)))]\\
& = E_0 + \sum_{j=1}^g (E_{2j-1} + E_{2j}
- 2\lam_j) - 2\pa_x^2 \ln
[\theta(\uxi_{P_0} + \ua_{P_0}
 (\hat \lam_0^\beta (x)) + \ual_{P_0}
(\hulam^\beta (x)))],
\lb{3.70}
\end{split}
\end{align}
where
$\lam_j \in [E_{2j-1}, E_{2j}]$,
$1 \leq j \leq g$
are determined from
\begin{equation}
\ome_{P_\infty,0}^{(2)}  =
 -[2 R_{2g+1}^{1/2} (.)]^{-1} \prod_{j=1}^g
(\ti \pi -
\lam_j) d\ti \pi \underset{\zeta\to 0}{=}
[\zeta^{-2} + 0(1)
] \, d\zeta \mbox{ near } P_\infty
\lb{3.71}
\end{equation}
and the second equality in \eqref{3.70} is
a consequence of the
equivalence $\calD_{P_\infty \humu(x)}
\sim \calD_{\hat \lam_0^\beta (x)
\hulam^\beta (x)}$, i.e.,
\begin{equation}
\ua_{P_0} (P_\infty) + \ual_{P_0} (\humu(x))
= \ua_{P_0} (\hat
\lam_0^\beta (x)) + \ual_{P_0}
 (\hulam^\beta (x)), \; x \in \bbR.
\lb{3.72}
\end{equation}

\section{The Time-Dependent Formalism}
\setcounter{equation}{0}
\setcounter{thm}{0}

In this section we construct algebro-geometric
 solutions of the KdV
hierarchy corresponding to $g$-gap initial
values on the basis of a
suitable time-dependent generalization of
the polynomial approach
developed in Chapters 2 and 3.  Even though
the final results
\eqref{4.55}--\eqref{4.59} are well-known, in
fact, classical by now, the
approach presented in this section, based on
the fundamental meromorphic
function $\phi (P,x,t)$ in \eqref{4.15}, merits
attention since it easily
extends to general $1+1$-dimensional completely
integrable systems such
as the AKNS and Toda hierarchies.  (The
corresponding approach to the
Toda and Kac-van Moerbeke lattices is presented
in detail in \cite{5}.)
The results \eqref{4.40}--\eqref{4.54} in
connection with the general
$\beta$-boundary condition in \eqref{3.44} and
our strategy of proof of
the theta function representation of the
BA function
$\psi(P,x,x_0,t,t_0)$ in \eqref{4.55}, based
on \eqref{4.15} and
\eqref{4.16}, are new.

Our starting point will be a $g$-gap solution
$V^{(0)}$ of the stationary
$\kdv_g$ equation,
\begin{equation}
V^{(0)} (x) = E_0 + \sum_{j=1}^g [E_{2j-1}
+E_{2j} - 2\mu_j^\bze (x)],\;
\mu_j^\bze (x) \in [E_{2j-1}, E_{2j}],
\quad 1 \leq j \leq g
\lb{4.1}
\end{equation}
satisfying
\begin{equation}
\sum_{\ell =0}^g c_{g-\ell} f_{\ell+1, x} =0,
\quad c_0 =1,
\lb{4.2}
\end{equation}
where $f_{\ell +1}$ are given by \eqref{2.4}
with $V=V^\bze$.  Our
principal aim then is to construct the
KdV flow
\begin{equation}
\kdv_r (V) =0,\; V(x,t_0) = V^\bze (x),
\quad x\in\bbR
\lb{4.3}
\end{equation}
for some $r\in\bbN_0$.  In terms of Lax
operators this amounts to solving
\begin{equation}
\dfrac{d}{dt} L(t) -[P_{2r+1} (t), L(t) ] =0,
\; t\in\bbR,\; [P_{2g+1}
(t_0), L(t_0) ] =0.
\lb{4.4}
\end{equation}
As a consequence one then obtains that
\begin{equation}
[P_{2g+1} (t), L(t)]=0, \quad t\in\bbR,
\lb{4.5}
\end{equation}
\begin{equation}
-P_{2g+1} (t)^2 = R_{2g+1} (L(t)) =
\prod_{n=0}^{2g} (L(t) - E_n), \quad
t\in\bbR
\lb{4.6}
\end{equation}
since the $\kdv_r$ flows are isospectral
deformations of $L(t_0)$.  In
this paper we shall base the explicit
solution of \eqref{4.3} not
directly on \eqref{4.4} and \eqref{4.5} but
instead take the following
equations as our point of departure,
\begin{align}
& V_t = - \dfrac12 \hat F_{r,xxx} +
2(V-z) \hat F_{r,x} + V_x \hat F_r,
\quad (x,t) \in\bbR^2,
\lb{4.7}\\
& F_{g,xx} F_g - \dfrac12 F_{g,x}^2 -
2(V-z) F_g^2 = 2R_{2g+1}(z), \quad
(x,t) \in\bbR^2,
\lb{4.8}
\end{align}
where
\begin{equation}
F_g(z,x,t) = \prod_{j=1}^g [z-\mu_j (x,t)]
\lb{4.9}
\end{equation}
(cf.\ \eqref{2.14}, \eqref{2.18}, and
\eqref{2.23}).  In order to stress
the fact that the integration constants
$c_\ell$ used in $F_g$ and $F_r$
(cf.\ \eqref{2.10}, \eqref{2.14}) in general
can differ from each other,
we explicitly employ the notation $F_g$,
$G_{g-1}$, $H_{g+1}$, $K_{g+1}^\beta$, etc.\
 and $\hat F_r$, $\hat
G_{r-1}$, $\hat H_{r+1}$, $\hat K_{r+1}^\beta$,
etc.\ throughout this
section. Similarly to
\eqref{3.5}--\eqref{3.8}, \eqref{3.11}, and
\eqref{3.12} we have
\begin{equation}
\mu_{j,x}(x,t) =-2iR_{2g+1}^{1/2}
 (\hat \mu_j (x,t))
\prod^g_{\substack{k=1\\ k\neq j}}
[ \mu_j (x,t) - \mu_k (x,t)]^{-1},\;
1 \leq j \leq g,\; (x,t) \in\bbR^2,
\lb{4.10}
\end{equation}
\begin{equation}
\{ \hat \mu_j (x_0, t)\}_{1\leq j \leq g}
\subset K_g,
\; \ti\pi (\hat
\mu_j (x_0, t)) =
 \mu_j (x_0, t) \in [E_{2j-1}, E_{2j}], \;
1 \leq j \leq g,\; t\in\bbR,
\lb{4.11}
\end{equation}
\begin{equation}
G_{g-1} (z,x,t) = - \dfrac12 F_{g,x} (z,x,t)
= -i\sum_{j=1}^g R_{2g+1}^{1/2}(\hat \mu_j (x,t))
 \prod^g_{\substack{k=1\\ k\neq j}}
\left( \dfrac{z-\mu_k(x,t)}
{\mu_j (x,t) - \mu_k (x,t)}
\right),
\lb{4.12}
\end{equation}
\begin{align}
& R_{2g+1} (z) + G_{g-1} (z,x,t)^2
= F_g (z,x,t) H_{g+1} (z,x,t),
\lb{4.13}\\
& H_{g+1} (z,x,t) = \prod_{\ell =0}^g
[ z-\nu_\ell (x,t)].
\lb{4.14}
\end{align}
In analogy to \eqref{3.14} and \eqref{3.17}
one then considers the
meromorphic function $\phi (P,x,t)$ on
$K_g$,
\begin{equation}
\phi (P,x,t)  = \dfrac{i R_{2g+1}^{1/2} (P)
- G_{g-1} (\ti \pi (P),
x,t)}{F_g (\ti \pi (P), x,t)}
 = \dfrac{-H_{g+1}}{iR_{2g+1}^{1/2} (P) +
G_{g-1} (\ti \pi (P), x,t)},
\quad (x,t) \in\bbR^2
\lb{4.15}
\end{equation}
and the $t$-dependent BA function
$\psi(P,x,x_0, t, t_0)$, meromorphic on
$K_g \bs \{ P_\infty\}$,
\begin{multline}
\psi (P,x,x_0, t,t_0) =
\exp \big\{ \int_{t_0}^t \, ds [ \hat F_r (z,x_0,
s) \phi (P,x_0,s) + \hat G_{r-1} (z,x_0, s)]
+ \int_{x_0}^x \, dy \phi
(P,y,t)\big\},\\
(x,x_0, t,t_0) \in\bbR^4.
\lb{4.16}
\end{multline}

\begin{lem}\lb{l4.1}
Let $P = (z,\sig R_{2g+1} (z)^{1/2})
= (\ti \pi (P),R_{2g+1}^{1/2} (P))\in K_g
\bs \{ P_\infty\}$, $(z,x,x_0, t,t_0)
\in\bbC \times \bbR^4$,
$r\in\bbN_0$.  Then
\begin{alignat}{2}
& \mbox{(i).}\quad && V(x,t) =
E_0+ \sum_{j=1}^g
[E_{2j-1} + E_{2j} - 2\mu_j (x,t)].
 \hspace*{7cm}
\lb{4.17}\\
& \mbox{(ii).}
&& \phi (P,x,t) \mbox{ satisfies } \notag\\
&&& \phi_x (P,x,t) + \phi (P,x,t)^2 =
V(x,t) -z,
\lb{4.18}\\
&&& \phi_t (P,x,t) =
\pa_x [\hat F_r (z,x,t) \phi (P,x,t) +
\hat G_{r-1}
(z,x,t) ].
\lb{4.19}\\
& \mbox{(iii).} &&
\psi (P,x,x_0, t,t_0) \mbox{ satisfies } \notag\\
&&& - \psi_{xx} (P,x,x_0, t,t_0) +
[ V(x,t) -z] \psi (P,x,x_0, t,t_0) =0,
\lb{4.20}\\
&&& \psi_t (P,x,x_0,t,t_0) =
\hat F_r (z,x,t) \psi_x
(P,x,x_0,t,t_0)
+ \hat G_{r-1} (z,x,t) \psi (P,x,x_0,t,t_0).
\lb{4.21}\\
& \mbox{(iv).} && \phi (P,x,t) \phi (P^*, x, t)
= H_{g+1} (z,x,t) / F_g
(z,x,t).
\lb{4.22}\\
& \mbox{(v).} && \phi (P,x,t) + \phi (P^*,x,t)
= -2G_{g-1} (z,x,t) / F_g
(z,x,t)
= F_{g,x} (z,x,t) / F_g (z,x,t).
\lb{4.23}\\
& \mbox{(vi).} && \phi (P,x,t) -
 \phi (P^*, x, t)
= 2i R_{2g+1}^{1/2}(P)
/ F_g (z,x,t).
\lb{4.24}
\end{alignat}
\end{lem}

\begin{proof}
The proof of (i), \eqref{4.18}, \eqref{4.20},
and (iv)--(vi) is analogous
to that in Lemma~\ref{l3.1}.  In order to
prove \eqref{4.19} one can
argue as follows.  By \eqref{4.7} and
\eqref{4.18},
$$
\pa_t (\phi_x + \phi^2) = \phi_{tx} +
2\phi\phi_t = V_t =(\hat F_r \phi +
\hat G_{r-1})_{xx} + 2\phi (\hat F_r \phi
+ \hat G_{r-1})_x,
$$
which implies
$$
(\pa_x + 2\phi) [\phi_t - (\hat F_r \phi
+ \hat G_{r-1})_x]=0
\text{ and hence }
\phi_t = (\hat F_r \phi + \hat G_{r-1} )_x
+ C e^{-2 \int^x \, dy \phi},
$$
where $C$ is independent of $x$ (but may
depend on $P$ and $t$).  The
high-energy behavior of $\phi (P,x,t)$
derived from \eqref{4.15} yields
$\phi (P,x,t) \underset{z\to\infty}{=}
 \pm i (z)^{1/2} + 0(1)$,
$P\in\Pi_\pm$ uniformly in
$(x,t) \in\bbR^2$ and hence
$C=0$ proving \eqref{4.19}.
\eqref{4.21} then immediately
follows from \eqref{4.16} and
\eqref{4.19}.
\end{proof}

In analogy to  \eqref{3.27} we now introduce
\begin{equation}
\hat{H}_{r+1} (z,x,t) =
\dfrac12 \hat{F}_{r,xx} (z,x,t) - [V(x,t) -z]
\hat{F}_r (z,x,t).
\lb{4.29}
\end{equation}
{}From \eqref{4.7} and \eqref{4.8} one then
computes
\begin{equation}
\hat{H}_{r+1,x}  = \dfrac12 \hat{F}_{r,xxx}-
(V-z) \hat{F}_{r,x}
- V_x \hat{F}_r = -V_t -2(V-z) \hat G_{r-1}.
\lb{4.30}
\end{equation}
The $t$-dependence of $F_g$, $G_{g-1}$, and
$H_{g+1}$ is governed by

\begin{lem} \lb{l4.2}
Let $(z,x,t) \in \bbC\times \bbR^2$,
$r\in\bbN_0$.  Then
\begin{alignat}{2}
& \mbox{(i).}\qquad && F_{g,t} (z,x,t)
= 2[F_g (z,x,t)
\hat G_{r-1} (z,x,t) -
\hat F_r (z,x,t) G_{g-1} (z,x,t)].
\hspace*{3cm}\lb{4.31}\\
& \mbox{(ii).} && G_{g-1,t} (z,x,t) =
F_g (z,x,t) \hat H_{r+1} (z,x,t) -
\hat F_r (z,x,t) H_{g+1} (z,x,t).
\lb{4.32}\\
& \mbox{(iii).} && H_{g+1,t} (z,x,t) =
2 [ \hat H_{r+1} (z,x,t) G_{g-1}
(z,x,t) - H_{g+1} (z,x,t) \hat G_{r-1} (z,x,t)].
\lb{4.33}
\end{alignat}
\end{lem}

\begin{proof}
By \eqref{2.17}, \eqref{4.19}, and
\eqref{4.24},
\begin{align*}
& \phi_t (P) - \phi_t (P^*)=
-2i R_{2g+1}^{1/2} (P) F_g^{-2} F_{g,t}\\
& = \pa_x [\hat F_r (\phi (P)-\phi (P^*))]
= 2iR_{2g+1}^{1/2}(P) (F_g
\hat F_{r,x} - F_{g,x} \hat F_r) F_g^{-2},
\end{align*}
implying \eqref{4.31}.  Similarly,
by \eqref{2.17} and \eqref{4.29},
$$
G_{g-1,t} =-\textstyle\frac12 F_{g,tx}
= \hat F_r G_{g-1,x} - F_g \hat G_{r-1,x}
 =
F_g \hat H_{r+1} - \hat F_r H_{g+1}.
$$
Finally, \eqref{2.17},
\eqref{4.29}--\eqref{4.32} yield
\begin{align*}
& H_{g+1,t} = -G_{g-1,tx}-(V-z) F_{g,t}
-V_t F_g
= -F_{g,x} \hat H_{r+1} -F_g \hat H_{r+1,x}
+ \hat F_{r,x} H_{g+1} +
\hat F_r H_{g+1,x}\\
& \quad -2 (V-z) (F_g \hat G_{r-1}
-\hat F_r G_{g-1}) -V_t F_g
= 2 (G_{g-1}\hat H_{r+1}
-2\hat G_{r-1} H_{g+1} ). \qed
\end{align*}
\renewcommand{\qed}{}
\end{proof}

As a consequence of \eqref{4.31} one
obtains the following
time-dependence of $\mu_j (x,t)$.

\begin{cor}\lb{c4.3}
Let $(x,t) \in\bbR^2$, $r\in\bbN_0$.
Then
\begin{align}
\begin{split}
\mu_{j,t} (x,t) & =
-2i \hat F_r (\mu_j (x,t),x) R_{2g+1}^{1/2} (\hat
\mu_j (x,t)) \prod^g_{\substack{ k=1\\ k\neq j}}
[\mu_j (x,t) -\mu_k (x,t)]^{-1},\\
\hat \mu_j (x,t_0) & = \hat \mu_j^\bze (x),
\quad 1\leq j \leq g,
\lb{4.34}
\end{split}\\
\ti\pi (\hat\mu_j (x,t)) & =
\mu_j (x,t) \in[E_{2j-1}, E_{2j}],\quad
1\leq j \leq g.
\lb{4.35}
\end{align}
\end{cor}

\begin{proof}
Take $z=\mu_j (x,t)$ in \eqref{4.31}
and observe \eqref{4.1},
\eqref{4.3}, \eqref{4.9}, and
\begin{equation}
 R_{2g+1}^{1/2} (\hat\mu_j (x,t))
=iG_{g-1} (\mu_j (x,t),x,t),\;
 \hat\mu_j (x,t) = (\mu_j (x,t),
iG_{g-1} (\mu_j (x,t), x,t)),
\lb{4.36}
\end{equation}
the latter fact following from \eqref{4.12}
(as in \eqref{3.9}).
\end{proof}

One observes that the special case
$r=0$ (i.e., $\hat F_0 =1$) in
\eqref{4.34} is equivalent to \eqref{4.10},
\eqref{4.11}.

Next we record the remaining $t$-dependent
analogs of Lemma~\ref{l3.1}
(vii)--(ix).

\begin{lem} \lb{l4.4}
Let $P=(z,\sig R_{2g+1} (z)^{1/2})
= (\ti \pi (P),R_{2g+1}^{1/2} (P)) \in K_g \bs
\{ P_\infty\}$, $(z,x,x_0,t,t_0)
\in\bbC \times \bbR^4$, $r\in\bbN_0$.
Then
\begin{alignat}{2}
& \mbox{(i).}\qquad &&
\psi (P,x,x_0,t,t_0) \psi (P^*,
x,x_0,t,t_0) =
F_g (z,x,t) / F_g (z,x_0,t_0). \hspace*{4cm}
\lb{4.37}\\
& \mbox{(ii).} &&
\psi_x (P,x,x_0,t,t_0) \psi_x (P^*, x,x_0,t,t_0) =
H_{g+1} (z,x,t) / F_g (z,x_0,t_0).
\lb{4.38}\\
& \mbox{(iii).} && \psi(P,x,x_0,t,t_0) =
[F_g (z,x,t) / F_g
(z,x_0,t_0)]^{1/2} \bullet \notag\\
&&& \bullet \exp \big\{iR_{2g+1}^{1/2} (P)
 \big[\int_{t_0}^t \, ds
\hat F_r (z,x_0,s) F_g(z,x_0,s)^{-1} +
 \int_{x_0}^x \, dy F_g
(z,y,t)^{-1}\big]\big\}.
\lb{4.39}
\end{alignat}
\end{lem}

\begin{proof}
(i).  Combining \eqref{4.16}, \eqref{4.23},
and \eqref{4.31} yields
\begin{align*}
&\quad \psi (P,x,x_0,t,t_0)
 \psi (P^*,x,x_0,t,t_0)\\
& = \exp \big[ \int_{t_0}^t \, ds
F_{g,s}(z,x_0,s) F_g(z,x_0,s)^{-1} +
\int_{x_0}^x \, dy F_{g,y} (z,y,t)
F_g (z,y,t)^{-1} \big]\\
& = [F_g (z,x_0,t) / F_g (z,x_0,t_0)]
[F_g(z,x,t)/ F_g (z,x_0,t)]
= F_g (z,x,t) / F_g (z,x_0,t_0).
\end{align*}
(ii).  \eqref{4.22}, \eqref{4.37} and
$\psi_x=\phi \psi$ imply
\begin{align*}
&\quad \psi_x (P,x,x_0,t,t_0)
 \psi_x (P^*,x,x_0,t,t_0)\\
& = [H_{g+1} (z,x,t) / F_g (z,x,t)]
[F_g(z,x,t)/ F_g (z,x_0,t_0)]
= H_{g+1} (z,x,t) / F_g (z,x_0,t_0).
\end{align*}
(iii).  Follows from \eqref{4.15},
\eqref{4.16}, and
\eqref{4.31}.
\end{proof}

\begin{rem} \lb{r4.5}
We emphasize that instead of taking
\eqref{4.7} and \eqref{4.8} as our
starting point for solving \eqref{4.3},
and subsequently deriving the
first-order differential system \eqref{4.10},
\eqref{4.34}, one could
have
started directly with the system \eqref{4.10},
\eqref{4.34} and derived
\eqref{4.7}, \eqref{4.8} and the remaining
facts of this section (cf.\
\cite{5}).
\end{rem}

Next, we turn to the $t$-dependent analog of
\eqref{3.50}--\eqref{3.63}
and start by introducing
\begin{equation}
K_{g+1}^\beta (z,x,t) = H_{g+1} (z,x,t)
-2\beta G_{g-1} (z,x,t) +
\beta^2 F_g(z,x,t)
= \prod_{\ell=0}^g [z-\lam_\ell^\beta (x,t)],
\quad \beta \in\bbR,
\lb{4.40}
\end{equation}
with
\begin{equation}
H_{g+1} (z,x,t) = K_{g+1}^0 (z,x,t), \;
\nu_\ell (x,t) = \lam_\ell^0
(x,t),\quad 0\leq \ell\leq g.
\lb{4.41}
\end{equation}
One then verifies in analogy to
\eqref{3.57}--\eqref{3.62} that
\begin{align}
\begin{split}
\phi (P,x,t) + \beta & =
\dfrac{iR_{2g+1}^{1/2}(P) - G_{g-1} (\ti\pi (P),
x,t) + \beta F_g(\ti \pi (P), x,t)}
{F_g (\ti \pi (P), x,t)}\\
& = \dfrac{-K_{g+1}^\beta (\ti \pi (P),x,t)}
{ iR_{2g+1}^{1/2} (P) +
G_{g-1} (\ti\pi (P),x,t) -
\beta F_g (\ti \pi (P),x,t)},
\lb{4.42}
\end{split}
\end{align}
\begin{align}
& R_{2g+1} (z) + [G_{g-1} (z,x,t)
-\beta F_g (z,x,t)]^2 = F_g (z,x,t)
K_{g+1}^\beta (z,x,t),
\lb{4.43}\\
& [\phi(P,x,t) + \beta][\phi (P^*,x,t)+
\beta] = K_{g+1}^\beta (z,x,t) /
F_g (z,x,t),
\lb{4.44}
\end{align}
\begin{align}
\begin{split}
& [\psi_x (P,x,x_0,t,t_0) +
\beta \psi (P,x,x_0,t,t_0) ][\psi_x
(P^*,x,x_0,t,t_0) +
\beta \psi (P^*,x,x_0,t,t_0)]\\
& = K_{g+1}^\beta (z,x,t)
/ F_g (z,x_0,t_0),
\lb{4.45}
\end{split}\\
& (\phi (.,x,t) + \beta ) =
\calD_{\hat \lam_0^\beta (x,t) \hulam^\beta
(x,t) } - \calD_{P_\infty \humu (x,t)},
\lb{4.46}
\end{align}
with
\begin{align}
\begin{split}
& R_{2g+1}^{1/2}
 (\hat \lam_\ell^\beta (x,t)) =- iG_{g-1}
(\lam_\ell^\beta
(x,t), x, t) +
i\beta F_g (\lam_\ell^\beta (x,t),x,t),\\
& \hat \lam_\ell^\beta (x,t) =
 (\lam_\ell^\beta (x,t), -iG_{g-1}
(\lam_\ell^\beta (x,t), x, t) +
i\beta F_g (\lam_\ell^\beta
(x,t),x,t)),\quad 0 \leq \ell \leq g.
\lb{4.47}
\end{split}
\end{align}
Eq.~\eqref{4.40} and Lemma~\ref{l4.2}
then yield
\begin{align}
\begin{split}
K_{g+1,t}^\beta (z,x,t) & =
2 \{ \hat K_{r+1}^\beta (z,x,t) [G_{g-1}
(z,x,t) - \beta F_g (z,x,t)]\\
& \quad - K_{g+1}^\beta (z,x,t)
[\hat G_{r-1} (z,x,t) - \beta \hat F_r
(z,x,t)]\}
\lb{4.48}
\end{split}
\end{align}
and in analogy to Corollary~\ref{c4.3}
one obtains from \eqref{4.48},
\begin{align}
\begin{split}
& \lam_{\ell,t}^\beta (x,t) =
-2i\hat K_{r+1}^\beta (\lam_\ell^\beta
(x,t), x,t) R_{2g+1}^{1/2}
 (\hat \lam_\ell^\beta (x,t))
\prod^g_{\substack{m=0\\ m\neq \ell}}
 [\lam_\ell^\beta (x,t)
-\lam_m^\beta (x,t)]^{-1},\\
& \hat \lam_\ell^\beta (x,t_0)
= \hat \lam_\ell^{\beta, (0)} (x), \quad
0 \leq \ell \leq g,\; (x,t) \in\bbR^2,
\lb{4.49}
\end{split}
\end{align}
\begin{equation}
\ti\pi (\hat \lam_0^\beta (x,t))
= \lam_0^\beta (x,t) \leq E_0,\; \ti \pi
(\hat \lam_\ell^\beta (x,t)) =
\lam_\ell^\beta (x,t) \in [E_{2\ell -1},
E_{2\ell}],\quad
(x,t) \in\bbR^2,
\lb{4.50}
\end{equation}
where $\{ \lam_\ell^{\beta,
(0)} (y) \}_{0 \leq \ell \leq g}$ are the
corresponding eigenvalues of
$H_y^{\beta, (0)}$ (cf.\ \eqref{3.44},
\eqref{3.54}, and \eqref{3.56}) associated
with the initial value
$V^\bze (x)$ in \eqref{4.1}.

In an analogous fashion one can analyze
the behavior of $\lam_\ell^\beta
(x,t)$ as a function of $\beta \in\bbR$.
In fact, \eqref{4.40} yields
\begin{align}
& \dfrac{\pa}{\pa\beta} K_{g+1}^\beta (z,x,t)
= -2 [G_{g-1} (z,x,t)
-\beta F_g (z,x,t)]
\lb{4.51}\\
\intertext{and hence}
\begin{split}
& \dfrac{\pa}{\pa\beta}
K_{g+1}^\beta (z,x,t) \big|_{z=\lam_\ell^\beta
(x,t)} =-\big[ \dfrac{\pa}{\pa\beta}
\lam_\ell^\beta (x,t) \big]
\prod^g_{\substack{m=0\\ m\neq \ell}}
[\lam_\ell^\beta (x,t) -
\lam_m^\beta (x,t)]\\
& = -2 [G_{g-1} (\lam_\ell^\beta (x,t), x,t)
- \beta F_g (\lam_\ell^\beta
(x,t), x,t)] =-2iR_{2g+1}^{1/2}
 (\hat \lam_\ell^\beta (x,t))
\lb{4.52}
\end{split}
\end{align}
by \eqref{4.47}. This implies for
$(\beta,x,t)\in\bbR^3$,
\begin{equation}
 \dfrac{\pa}{\pa\beta} \lam_\ell^\beta (x,t)
= 2i R_{2g+1}^{1/2} (\hat
\lam_\ell^\beta (x,t))
\prod^g_{\substack{m=0\\ m\neq \ell}}
[\lam_\ell^\beta (x,t) -
\lam_m^\beta (x,t)]^{-1},\quad
0 \leq \ell \leq g.
\lb{4.53}
\end{equation}

As in Section~3 we conclude with the
$\theta$-function representation of
$\phi (P,x,t)$, $\psi(P,x,x_0,t,t_0)$,
and $V(x,t)$.

\begin{thm} \lb{t4.6}
Let $P=(z,\sig R_{2g+1} (z)^{1/2})
 \in K_g \bs \{ P_\infty\}$,
$(z,x,x_0,t,t_0) \in\bbC \times \bbR^4$,
$P_0=(E_0,0)$.  Then
\begin{multline}
\phi (P,x,t) = -\beta +
\dfrac{\theta (\uxi_{P_0} - \ua_{P_0} (P_\infty)
+ \ual_{P_0} (\humu (x,t)))}{\theta (\uxi_{P_0}
- \ua_{P_0} (P_\infty)
+\ual_{P_0} (\hulam^\beta (x,t)))} \bullet\\
\bullet \dfrac{
\theta (\uxi_{P_0} -\ua_{P_0} (P)
+ \ual_{P_0}
(\hulam^\beta (x,t)))}{
\theta (\uxi_{P_0} -\ua_{P_0} (P)
+ \ual_{P_0}
(\humu (x,t)))}
\exp \big[-\int_{P_0}^P \ome^\bth_{P_\infty,
  \hat
\lam_0^\beta (x,t)} \big]
\lb{4.54}
\end{multline}
and
\begin{multline}
\psi (P,x,x_0,t,t_0) =
\dfrac{\theta (\uxi_{P_0} - \ua_{P_0} (P)
+ \ual_{P_0} (\humu (x,t)))}
{\theta (\uxi_{P_0} - \ua_{P_0} (P_\infty)
+\ual_{P_0} (\humu (x,t)))} \bullet\\
\bullet \dfrac{
\theta (\uxi_{P_0} -\ua_{P_0} (P_\infty)
+ \ual_{P_0}
(\humu (x_0,t_0)))}{
\theta (\uxi_{P_0} -\ua_{P_0} (P)
+ \ual_{P_0}
(\humu (x_0,t_0)))}
 \exp \big[-i (x-x_0) \int_{P_0}^P
\ome_{P_\infty,0}^\btwo -
i (t-t_0) \int_{P_0}^P \Ome_{P_\infty,2r}^\btwo
\big],
\lb{4.55}
\end{multline}
where (cf.\ \eqref{a.34})
\begin{align}
& \Ome_{P_\infty,2r}^\btwo  =
\sum_{s=0}^r c_{r-s} (2s+1) \ome_{P_\infty,
2s}^\btwo,
\lb{4.56}\\
& \ual_{P_0} (\humu (x,t)) =
\ual_{P_0} (\humu (x_0,t_0)) +
\dfrac{\uu_0^\btwo}{2\pi} (x-x_0)
+ \dfrac{\uu_{2r}^\btwo}{2\pi}
(t-t_0),
\lb{4.57}\\
& \uu_{2r}^\btwo = (U_{2r,1}^\btwo,\ldots,
U_{2r,g}^\btwo),\;
U_{2r,j}^\btwo = \int_{b_j
} \Ome_{P_\infty,2r}^\btwo, \quad 1 \leq j \leq
g.
\lb{4.58}
\end{align}
The Its-Matveev formula (\cite{26},
\cite{4}, Ch.~3, \cite{38}, Ch.~II)
for
$V(x,t)$ reads (cf.\ \eqref{3.70})
\begin{equation}
V(x,t) = E_0+ \sum_{j=1}^g (E_{2j-1}
+ E_{2j} - 2\lam _j)
 - 2 \pa_x^2 \ln [\theta (\uxi_{P_0}
-\ua_{P_0} (P_\infty) +
\ual_{P_0} (\humu (x,t)))].
\lb{4.59}
\end{equation}
\end{thm}

\begin{proof}[Sketch of Proof]
Since \eqref{4.54} follows directly from
\eqref{4.46} and \eqref{a.37},
and \eqref{4.59} can be inferred from
\eqref{4.55} and
\eqref{4.20} upon expanding all quantities
in \eqref{4.20} near
$P_\infty$ in a well-known manner, we
first concentrate on the proof of
\eqref{4.55}.  Let $\psi (P,x,x_0,t,t_0)$
be defined as in \eqref{4.16}
and denote the right-hand-side of
\eqref{4.55} by $\Psi (P,x,x_0,t,t_0)$.
In order to prove that $\psi = \Psi$, one
first observes from
\eqref{4.10} and \eqref{4.34} that
\begin{equation}
\hat F_r (\ti\pi (P), x_0,s) \phi (P,x_0,s)
= \textstyle\frac{\pa}{\pa s} \ln
[\mu_j (x_0,s) - \ti\pi (P) ] + 0(1)
\mbox{ for } P \mbox{ near } \hat\mu_j (x_0,s)
\lb{4.60}
\end{equation}
and
\begin{equation}
\phi (P,y,t) = \textstyle\frac{\pa}{\pa y}
 \ln [\mu_j(y,t) -\ti\pi (P)] + 0(1)
\mbox{ for } P \mbox{ near } \hat\mu_j (y,t).
\lb{4.61}
\end{equation}
Hence
\begin{align}
\begin{split}
& \exp \big\{ \int_{t_0}^t \, ds
[ \textstyle\frac{\pa}{\pa s} \ln (\mu_j
(x_0,s) - \ti\pi (P)) + 0(1)]\big\}\\
& = \begin{cases}
[\mu_j (x_0,t) - \ti \pi (P)] 0(1)
& \mbox{ for } P \mbox{ near } \hat
\mu_j (x_0,t) \neq \hat \mu_j (x_0,t_0)\\
0(1) & \mbox{ for } P \mbox{ near }
 \hat \mu_j (x_0,t) = \hat \mu_j
(x_0,t_0)\\{}
[\mu_j(x_0,t_0) - \ti \pi (P) ]^{-1} 0(1)
& \mbox{ for } P \mbox{ near }
\hat \mu_j (x_0, t_0)
 \neq \hat \mu_j (x_0, t)
\end{cases}
\lb{4.62}
\end{split}\\
\intertext{and}
\begin{split}
& \exp \big\{ \int_{x_0}^x \, dy
[ \textstyle\frac{\pa}{\pa y} \ln (\mu_j
(y,t) - \ti\pi (P)) + 0(1) ] \big\} \\
& = \begin{cases}
[\mu_j (x,t) - \ti \pi (P)] 0(1)
& \mbox{ for } P \mbox{ near } \hat
\mu_j (x,t) \neq \hat \mu_j (x_0,t)\\
0(1) & \mbox{ for } P \mbox{ near }
 \hat \mu_j (x,t) = \hat \mu_j
(x_0,t)\\{}
[\mu_j(x_0,t) - \ti \pi (P) ]^{-1} 0(1)
& \mbox{ for } P \mbox{ near }
\hat \mu_j (x_0, t) \neq \hat \mu_j (x, t)
\end{cases},
\lb{4.63}
\end{split}
\end{align}
where $0(1) \neq 0$ in \eqref{4.62} and
\eqref{4.63}.  Consequently, all
zeros and poles of $\psi$ and $\Psi$ on
$K_g \bs \{ P_\infty\}$ are
simple and coincide.  By an application of
the Riemann-Roch theorem it
remains to identify the essential singularity
of $\psi$ and $\Psi$ at
$P_\infty$.  For that purpose we first
recall the known fact that the
diagonal Green's function $G(z,x,x,t)$ of
$H(t)$ satisfies
\begin{equation}
G(z,x,x,t) \underset{\zeta \to 0}{=}
 (i/2) \zeta \sum_{j=0}^\infty \ti
f_j (x,t) \zeta^{2j},\quad \zeta =1/ \sqrt z,
\lb{4.64}
\end{equation}
with $\ti f_j(x,t)$ the homogeneous
coefficients as introduced in the
context of \eqref{2.12} satisfying the
recursion \eqref{2.4} for all
$j\in\bbN$.
Combining
\begin{equation}
G(z,x,x,t) = \dfrac{iF_g (z,x,t)}
{2R_{2g+1} (z)^{1/2}}
\lb{4.66}
\end{equation}
(cf.\ \eqref{3.34}), \eqref{4.15},
\eqref{4.16}, and \eqref{4.64}
then yields
\begin{equation}
 \int_{x_0}^x \, dy \phi (P,y,t)
 \underset{\zeta\to 0}{=} \int_{x_0}^x
\, dy \dfrac{iR_{2g+1}^{1/2} (P)}
{F_g (\ti \pi (P), y,t)} + 0 (\zeta^2)
 \underset{\zeta \to 0}{=} i(x-x_0)
[\zeta^{-1} + 0(1)],
\lb{4.67}
\end{equation}
which coincides with the singularity at
$P_\infty$ of the $x$-dependent
term in the exponent of \eqref{4.55}
taking into account \eqref{3.71}.
Finally, in order to identify the
$t$-dependent essential singularity of
$\psi$ and $\Psi$, we may allude to
\eqref{2.20} and, without loss of
generality, consider the homogeneous case
where $c_0 =1$, $c_q =0$,
$1\leq q \leq r$.  Invoking \eqref{4.31}
then yields from \eqref{4.15}
and \eqref{4.66}
\begin{align}
\begin{split}
& \int_{t_0}^t \, ds [\ti F_r (z,x_0,s)
\phi (P,x_0,s) + \ti G_{r-1}
(z,x_0,s)]\\
& = \int_{t_0}^t \,
ds \big\{ \ti F_r (z,x_0,s) iR_{2g+1}^{1/2} (P)
F_g(z,x_0,s)^{-1} +
\textstyle\frac12 \textstyle\frac{\pa}{\pa s} \ln
 [F_g(z,x_0,s)]
\big\}\\
& \underset{\zeta \to 0}{=}
-\textstyle\frac12 \int_{t_0}^t \, ds \ti F_r
 (z,x_0,
s) G(z,x_0,x_0,s)^{-1} +0(1),\quad \zeta
= 1/ \sqrt z.
\lb{4.68}
\end{split}
\end{align}
Comparing \eqref{2.14} (in the homogeneous
case) and \eqref{4.64} implies
\begin{align}
& - \textstyle\frac12 \ti F_r (z,x_0,s)
 G(z,x_0,x_0,s)^{-1} \underset{\zeta \to
0}{=} i\zeta^{-2r-1} +0(1)
\lb{4.69}\\
\intertext{and hence}
& \int_{t_0}^t \, ds [ \ti F_r (z,x_0,s)
 \phi (P,x_0,s) + \ti G_{r-1}
(z,x_0,s)]
\underset{\zeta \to 0}{=} i (t-t_0)
[\zeta^{-2r-1} +0(1)],
\lb{4.70}
\end{align}
completing the proof of \eqref{4.55}.
The linearity of the Abel map with
respect to $x$ and $t$ in \eqref{4.57}
then follows by a standard
argument considering the differential
$\Ome (x,x_0,t,t_0) = d\ln \psi
(.,x,x_0,t,t_0)$.
\end{proof}

\section{General Trace Formulas} \lb{s5}
\setcounter{equation}{0}
\setcounter{thm}{0}

Following a recent series of papers on new
trace formulas for \schro
operators \cite{16}--\cite{19},
\cite{22}--\cite{24}, \cite{39}, we first
discuss appropriate Krein spectral shift
functions, the key tool for
general higher-order trace formulas.
Subsequently, we develop a new
method for deriving small-time heat
kernel (respectively high-energy
resolvent) expansion coefficients
associated with the general
$\beta$-boundary conditions in \eqref{5.3}.
Interest in these types of
trace formulas stems from their crucial role
in the solution of inverse
spectral problems.

Unlike Sections 3 and 4, where we focused
on the special case of
stationary finite-gap solutions of the
KdV hierarchy (the natural
extension of solitons as reflectionless
potentials), we now turn to the
general situation and consider potentials
of the type
\begin{equation}
V\in C^\infty (\bbR),\; V(x) \geq c,\;
x\in\bbR,\; V \mbox{ real-valued.}
\lb{5.1}
\end{equation}
As in Section 3 we introduce the
differential expression $\tau = -
\textstyle\frac{d^2}{dx^2} + V(x)$, $x\in\bbR$
and define the self-adjoint
operators $H$ and $H_{x_0}^\beta$ in $L^2(\bbR)$,
\begin{equation}
Hf = \tau f,\;
f\in\calD (H) = \{ g\in L^2 (\bbR) | g,g'
\in AC_{\loc} (\bbR);\, \tau
g\in L^2 (\bbR)\}
\lb{5.2}
\end{equation}
and for $\beta \in \bbR \cup \{ \infty\},
\; x_0\in \bbR$,
\begin{align}
\begin{split}
H_{x_0}^\beta f = \tau f,\;
f\in \calD (H_{x_0}^\beta ) =
& \{ f\in L^2 (\bbR) | g,g' \in AC ([x_0,
\pm R]) \mbox{ for all } R> 0,\\
& \quad \lim\limits_{\eps \downarrow 0
} [g' (x_0 \pm \eps )+ \beta g(x_0
\pm \eps)] =0,\quad \tau g \in L^2 (\bbR)\},
\lb{5.3}
\end{split}
\end{align}
with $H_{x_0}^\infty = H_{x_0}^D
\;(H_{x_0}^0 = H_{x_0}^N)$ the
corresponding Dirichlet (Neumann) \schro
operator.  If $G(z,x,x')$
denotes the Green's function of $H$ (as
in \eqref{3.31}, \eqref{3.32}),
formulas \eqref{3.47}--\eqref{3.49} for
the resolvent of $H_{x_0}^\beta$
apply without change in the present general
situation.  In particular,
defining
\begin{equation}
\Gam^\beta (z,x) = \begin{cases}
(\beta + \pa_1)(\beta + \pa_2) G(z,x,x),
& \beta \in\bbR\\
G(z,x,x), & \beta = \infty
\end{cases}
\lb{5.4}
\end{equation}
(cf.\ the notation introduction in \eqref{3.38})
one computes for
$\beta \in \bbR \cup \{\infty\}$,
\begin{equation}
\tr [(H_x^\beta -z)^{-1} -(H-z)^{-1}] =
- \dfrac{d}{dz} \ln [\Gam^\beta
(z,x)],\;
z\in\bbC \bs \{ \sig (H_x^\beta ) \cup
\sig (H) \}.
\lb{5.5}
\end{equation}
Given hypothesis \eqref{5.1}, one can prove
the existence of asymptotic
expansions of the type
\begin{equation}
\tr [(H_x^\beta -z)^{-1}
-(H-z)^{-1}] \underset{z\to i\infty}{=}
\sum_{j=0}^\infty r_j^\beta (x) z^{-j},
\quad \beta\in\bbR \cup \{\infty\}
\lb{5.6}
\end{equation}
uniformly with respect to $x\in\bbR$
(cf.\ \cite{24}).  In particular,
one can derive the heat kernel expansion
\begin{equation}
\tr [e^{-\tau H_x^\infty} - e^{-\tau H}]
\underset{\tau\downarrow
0}{\sim}
 \sum_{j=0}^\infty s_j^\infty (x) \tau^j, \quad
x\in\bbR,
\lb{5.7}
\end{equation}
where
\begin{equation}
s_j^\infty (x) =(-1)^{j+1} (j!)^{-1}
 r_j^\infty (x),\quad j \in\bbN_0
\lb{5.8}
\end{equation}
and $s_j^\infty \; (r_j^\infty)$ are the
well-known invariants of the KdV
hierarchy.

In the special case of finite-gap
potentials the connection of
$\Gam^\beta (z,s)$ in \eqref{5.4} with
our polynomial approach in
Section~3 is clearly demonstrated by
\eqref{3.34} for $\beta = \infty$
and \eqref{3.53} for $\beta = \bbR$.

Before describing a new constructive (i.e.,
recursive) approach to the
coefficients $r_j^\beta (x)$, $\beta \in\bbR$,
we recall the definition of
Krein's spectral shift function \cite{30}
associated with the pair
$(H_x^\beta, H)$ (cf.\ \cite{19}, \cite{23},
\cite{24}).  The rank-one
resolvent difference of $H_x^\beta$ and $H$
(cf.\ \eqref{3.47},
\eqref{3.48}) is intimately connected with
the fact that for each
$x\in\bbR$, $\beta \in\bbR\cup\{\infty\}$,
\begin{equation}
\Gam^\beta (z,x) \mbox{ is Herglotz with
respect to } z
\lb{5.9}
\end{equation}
(i.e., a holomorphic map $\bbC_+ \to \bbC_+$,
where $\bbC_+ =\{z\in\bbC|
\iim
 (z) > 0 \}$).  The exponential Herglotz
representation for $\Gam^\beta
(z,x)$ (cf.\ \cite{3}) then reads for each
$x\in\bbR$,
\begin{multline}
\Gam^\beta (z,x) = \exp \big\{ c^\beta +
\int_\bbR [(\lam-z)^{-1} - \lam
(1+\lam^2)^{-1}][\xi^\beta (\lam,x) +
\del^\beta ] \, d\lam \big\},\\
c^\beta \in\bbR,\; \beta \in \bbR\cup \{\infty\},
 \; \del^\beta =
\begin{cases}
1, & \beta \in\bbR\\
0, & \beta = \infty
\end{cases},
\lb{5.10}
\end{multline}
where, by Fatou's lemma,
\begin{equation}
\xi^\beta (\lam,x) =
\pi^{-1} \lim\limits_{\eps \downarrow 0}
 \iim \{ \ln
[\beta + \pa_1)(\beta + \pa_2) G(\lam +
 i\eps, x, x) ] \} -
\del^\beta,\quad
\beta \in \bbR \cup \{ \infty\}
\lb{5.11}
\end{equation}
for a.e. $\lam \in\bbR$.  Moreover,
\begin{align}
\begin{split}
& -1 \leq \xi^\beta (\lam,x) \leq 0, \;
\xi^\beta (\lam,x) =0,\; \lam <
\iinf \sig (H_x^\beta ),\; \beta \in\bbR,\\
& 0 \leq \xi^\infty (\lam, x) \leq 1, \;
\xi^\infty (\lam, x) =0, \; \lam
< \iinf \sig (H)
\lb{5.12}
\end{split}
\end{align}
for a.e. $\lam \in\bbR$.  As a consequence,
one obtains (cf.\ \cite{39})
\begin{equation}
\tr [f(H_x^\beta)-f(H)] = \int_\bbR \,
d\lam f' (\lam) \xi^\beta (\lam,
x),\quad \beta\in\bbR \cup
\{\infty\},\; x\in\bbR
\lb{5.13}
\end{equation}
for any $f\in C^2 (\bbR)$ with $(1+
\lam^2) f^{(j)} \in L^2
((0,\infty))$, $j=1,2$ and for $f(\lam)
= (\lam -z)^{-1}$, $z\in\bbC \bs
[\iinf \sig (H_x^\beta), \infty)$.  In
particular, \eqref{5.13} holds for
traces of heat kernel and resolvent differences,
i.e., for any $\beta
\in\bbR\cup \{\infty\}$, $x\in\bbR$,
\begin{align}
& \tr [e^{-\tau H_x^\beta} - e^{-\tau H}]
= -\tau
\int_{e^\beta_{x,0}}^\infty \,
d\lam e^{-\tau \lam} \xi^\beta (\lam, x),
\quad \tau > 0,
\lb{5.14}\\
& \tr [(H_x^\beta -z)^{-1} - (H-z)^{-1} ]
 = -
\int_{e_{x,0}^\beta}^\infty
\, d\lam (\lam -z)^{-2} \xi^\beta (\lam, x),
\quad z\in\bbC \bs \{ \sig
(H_x^\beta ) \cup \sig (H)\},
\lb{5.15}
\end{align}
where
\begin{equation}
e_{x,0}^\beta = \left\{\begin{array}{ll}
\iinf \sig (H_x^\beta), & \beta \in\bbR\\
\iinf \sig (H), & \beta = \infty
\end{array}\right..
\lb{5.16}
\end{equation}

Returning to a recursive approach for the
expansion coefficients
$r_j^\beta (x)$ in \eqref{5.6} we first
consider the expansion
\begin{equation}
\Gam^\beta (z,x)
\underset{z\to i\infty}{=} (i/2)
\sum_{j=-\del^\beta}^\infty \gam_j^\beta (x)
 z^{-j -1/2}, \quad \beta
\in\bbR \cup \{ \infty\}.
\lb{5.17}
\end{equation}
(A comparison of \eqref{5.17} and
\eqref{4.64} reveals that
$\gam_j^\infty (x) = \ti f_j (x)$,
$j\in\bbN_0$ in the case $\beta =
\infty$.)  In order to obtain a recursion
relation for $\gam_j^\beta (x)$
one can use the following result.

\begin{lem} \lb{l5.1}
Let $z\in\bbC \bs \sig (H)$, $x\in\bbR$.\\
(i).  Assume $\beta \in\bbR$.  Then
$\Gam^\beta (z,x) = (\beta + \pa_1)
(\beta + \pa_2) G(z,x,x)$
satisfies
\begin{align}
\begin{split}
& 2 [V(x) - \beta^2 -z]
\Gam_{xx}^\beta (z,x)
\Gam^\beta (z,x) -[V(x) -
\beta^2 -z] \Gam_x^\beta (z,x)^2
-2V_x(x) \Gam_x^\beta (z,x)
\Gam^\beta (z,x) \\
& -4 \{ [ V(x) -z][V(x) -
\beta^2 -z] -\beta V_x(x) \} \Gam^\beta
(z,x)^2
 = - [V(x) -z-\beta^2]^3.
\lb{5.18}
\end{split}
\end{align}
(ii).  Assume $\beta=\infty$.  Then
$\Gam^\infty (z,x) = G(z,x,x)$
satisfies
\begin{equation}
\Gam_{xxx}^\infty (z,x) -4 [V(x) -z]
\Gam_x^\infty (z,x) -2V_x(x)
\Gam^\infty (z,x) =0
\lb{5.19}
\end{equation}
and
\begin{equation}
-2\Gam_{xx}^\infty (z,x) \Gam^\infty (z,x)
 +\Gam_x^\infty (z,x)^2 + 4
[V(x) -z] \Gam^\infty (z,x)^2 =1.
\lb{5.20}
\end{equation}
\end{lem}

While the results \eqref{5.19} and
\eqref{5.20} in the Dirichlet case
$\beta = \infty$ are well-known, see, e.g.,
\cite{14}, the result
\eqref{5.18} (with the exception of the
Neumann case $\beta=0$ which was
first presented in \cite{21}) is new.
Unfortunately, we have no
reasonably short derivation of the differential
 equation \eqref{5.18}.
It can be verified (not without tears) after
quite tedious though
straightforward calculations (we recommend
additional help in the form of
symbolic computations).

Insertion of the expansion \eqref{5.17} into
\eqref{5.18} and
\eqref{5.20} in Lemma~\ref{l5.1} yields

\begin{lem} \lb{l5.2}
The coefficients $\gam_j^\beta (x)$ in \eqref{5.17}
satisfy the following
recursion relation.\\
(i). Assume $\beta \in\bbR$.  Then
\begin{align}
\begin{split}
\gam_{-1}^\beta &= 1,\; \gam_0^\beta = \beta^2
- \textstyle\frac12 V,\;
\gam_1^\beta = \textstyle\frac12 \beta^2 V +
\textstyle\frac12 \beta V_x -
 \textstyle\frac18 V^2 +
\textstyle\frac18 V_{xx},\\
\gam_2^\beta & = -\textstyle\frac1{16} V^3 +
\textstyle\frac38 \beta^2 V^2 +
 \textstyle\frac3{16}
V_x(4\beta V + V_x) + \textstyle\frac18 V_{xx}
 (V-\beta^2)
- \textstyle\frac18 \beta V_{xxx} -
\textstyle\frac1{64} V_{xxxx},\\
\gam_{j+1}^\beta & =
\textstyle\frac18 \sum_{\ell =1}^j
[2 (V-\beta^2)
\gam_{\ell-1}^\beta \gam_{j-\ell, xx}^\beta
- (V-\beta^2) \gam_{\ell-1,
x}^\beta \gam_{j-\ell, x}^\beta\\
& \quad -4 \gam_\ell^\beta \gam_{j-\ell+1}^\beta
-4 V (V-\beta^2)
\gam_{\ell-1}^\beta \gam_{j-\ell}^\beta
-2V_x \gam_{\ell-1}^\beta
\gam_{j-\ell,x}^\beta +
\gam_{\ell-1}^\beta \gam_{j-\ell}^\beta]\\
&\quad + \textstyle\frac18 \sum_{\ell=0}^j
[\gam_{\ell,x}^\beta
\gam_{j-\ell,x}^\beta
-2\gam_\ell^\beta \gam_{j-\ell, xx}^\beta -4
(\beta^2 -2V)
 \gam_\ell^\beta \gam_{j-\ell}^\beta ],
\quad j\geq 2.
\lb{5.21}
\end{split}
\end{align}
(ii). Assume $\beta =\infty$.  Then
\begin{align}
\begin{split}
\gam_0^\infty & = 1, \; \gam_1^\infty
 = \textstyle\frac12 V,\\
\gam_{j+1}^\infty & = -
\textstyle\frac12 \sum_{\ell=1}^j
\gam_\ell^\infty
\gam_{j+1-\ell}^\infty +
\textstyle\frac12 \sum_{\ell=0}^j
[V\gam_\ell^\infty
\gam_{j-\ell}^\infty +
\textstyle\frac14 \gam_{\ell,x}^\infty
\gam_{j-\ell,x}^\infty -
\textstyle\frac12 \gam_{\ell,xx}^\infty
\gam_{j-\ell}^\infty],\quad j\geq 1.
\lb{5.22}
\end{split}
\end{align}
\end{lem}

The final result for $r_j^\beta (x)$ then reads

\begin{thm} \lb{t5.3}
The coefficients $r_j^\beta (x)$ in \eqref{5.6}
satisfy the following
recursion relations.\\
(i).  Assume $\beta \in\bbR$.  Then
\begin{equation}
 r_0^\beta (x)  =-\textstyle\frac12,\;
r_1^\beta (x) = \beta^2 -
 \textstyle\frac12
V(x),\;
 r_j^\beta (x)  = j \gam_{j-1}^\beta (x) -
\sum_{\ell=1}^{j-1}
\gam_{j-\ell -1}^\beta (x) r_\ell^\beta (x),
\quad j \geq 2.
\lb{5.23}
\end{equation}
(ii).  Assume $\beta = \infty$.  Then
\begin{equation}
r_0^\infty  = \textstyle\frac12, \; r_1^\infty (x)
= \textstyle\frac12 V(x),\;
r_j^\infty (x)  = j \gam_j^\infty (x) -
\sum_{\ell=1}^{j-1}
\gam_{j-\ell}^\infty (x) r_\ell^\infty (x),
\quad j\geq 2.
\lb{5.24}
\end{equation}
\end{thm}

\begin{proof}
It suffices to combine \eqref{5.5},
\eqref{5.6}, \eqref{5.17}, and the
following well-known fact on asymptotic
expansions:
$ F(z) \underset{|z|\to\infty}{=}
\sum_{j=1}^\infty c_j z^{-j}$
implies
$ \ln [1+F(z)] \underset{|z|\to\infty}{=}
\sum_{j=1}^\infty d_j z^{-j}$,
where
$d_1 = c_1$,
$d_j = c_j - \sum_{\ell=1}^{j-1} (\ell / j)
 c_{j-\ell} d_\ell$,
$j\geq 2$.
\end{proof}

Theorem~\ref{t5.3}~(i) has first been derived
(by using a different
strategy) in \cite{24}.  The current derivation,
based on the universal
differential equation \eqref{5.18}, is new.
Combined with \eqref{5.21},
Theorem~\ref{t5.3}~(i) yields the most
efficient algorithm to date for
computing $r_j^\beta (x)$, $\beta \in\bbR$.

The connection between $r_j^\beta (x)$ and
$\xi^\beta (\lam, x)$ is
illustrated in the following result.

\begin{thm} \lb{t5.4}
\cite{24} Let $e_{x,0}^\beta =
\iinf \sig (H_x^\beta)$, $\beta \in\bbR$,
$e_0^\infty = \iinf \sig (H)$.\\
(i). Assume $\beta\in\bbR$.  Then
\begin{equation}
r_j^\beta (x) = -
\textstyle\frac12 (e^\beta_{x,0} )^j -
\lim\limits_{z\to
i\infty} \int_{e_{x,0}^\beta}^\infty \,
d\lam z^{j+1} (\lam -z)^{-j-1}
j(-\lam)^{j-1} [ \textstyle\frac12 +
\xi^\beta (\lam,x)],
\; j \in \bbN.
\lb{5.25}
\end{equation}
(ii).  Assume $\beta = \infty$.  Then
\begin{equation}
r_j^\infty (x) =
\textstyle\frac12 (e_0^\infty )^j
+ \lim\limits_{z\to i\infty}
\int_{e_0^\infty}^\infty \,
d\lam z^{j+1} (\lam -z)^{-j-1} j
(-\lam)^{j-1} [ \textstyle\frac12
- \xi^\infty (\lam, x) ],\;
j \in\bbN.
\lb{5.26}
\end{equation}
\end{thm}

We conclude with an example that yields
the higher-order trace formulas
for periodic potentials which also applies
to the (quasi-periodic)
finite-gap potentials of Section~3.

\begin{exmp} \lb{e5.5}
Assume $V$ is periodic, i.e., for some
$\Ome > 0$, $V(x+ \Ome) = V(x)$
for all $x\in\bbR$ in addition to
\eqref{5.1}.  Then Floquet theory
implies
\begin{equation}
\sig (H) = \bigcup_{n=1}^\infty [E_{2(n-1)},
E_{2n-1}], \; E_0 < E_1 \leq
E_2 <E_3 \leq \cdots
\lb{5.27}
\end{equation}
(i).  Assume $\beta \in\bbR$.  Then
\begin{equation}
 \sig (H_x^\beta) =
\{ \lam_n^\beta (x) \}_{n\in\bbN_0}
\cup \sig (H),\;
 \lam_0^\beta (x) \leq E_0,\;
\lam_n^\beta (x) \in [E_{2n-1}, E_{2n}],
\quad n\in\bbN,
\lb{5.28}
\end{equation}
\begin{equation}
\xi^\beta (\lam, x) = \left\{\begin{array}{ll}
0, & \lam < \lam_0^\beta (x),\;
E_{2n-1} < \lam < \lam_n^\beta (x), \quad
n\in\bbN\\
-1, & \lam_0^\beta (x) < \lam < E_0, \;
\lam_n^\beta (x) < \lam < E_{2n},
\quad n\in\bbN\\
- \dfrac12, & E_{2(n-1)} < \lam < E_{2n-1},
\quad n\in\bbN
\end{array}\right..
\lb{5.29}
\end{equation}
Inserting \eqref{5.29} into \eqref{5.25} then
yields the higher-order
periodic trace formulas
\begin{equation}
r_j^\beta (x) = \textstyle\frac12 E_0^j -
\lam_0^\beta (x)^j + \textstyle\frac12
\sum_{n=1}^\infty [E_{2n-1}^j + E_{2n}^j -
2\lam_n^\beta (x)^j], \quad j
\in\bbN.
\lb{5.30}
\end{equation}
(ii).  Assume $\beta=\infty$.  Then
\begin{equation}
 \sig(H_x^\infty) = \{\mu_n (x)\}_{n\in\bbN}
\cup \sig (H),\;
\mu_n (x) \in [E_{2n-1}, E_{2n}],\quad n\in\bbN,
\lb{5.31}
\end{equation}
\begin{equation}
\xi^\infty (\lam, x) = \left\{\begin{array}{ll}
0, & \lam < E_0, \; \mu_n(x) < \lam < E_{2n},
\quad n\in\bbN\\
1, & E_{2n-1} < \lam < \mu_n (x),
\quad n\in\bbN\\
\dfrac12, & E_{2(n-1)} < \lam < E_{2n-1},
\quad n\in\bbN.
\end{array}\right..
\lb{5.32}
\end{equation}
Insertion of \eqref{5.32} into \eqref{5.26}
then yields
\begin{equation}
r_j^\infty (x) = \textstyle\frac12 E_0^j +
\textstyle\frac12 \sum_{n=1}^\infty
 [E_{2n-1}^j
+ E_{2n}^j -2\mu_n (x)^j],\quad j \in\bbN.
\lb{5.33}
\end{equation}
\end{exmp}

The results \eqref{5.29} and \eqref{5.32}
remain valid in the
algebro-geometric finite-gap situation
discussed in Section 3 where
\begin{equation}
E_{2n+1} = \lam_n^\beta (x) = E_{2n+2},
\quad n \geq g+1, \;
\beta \in\bbR\cup \{\infty\}.
\lb{5.34}
\end{equation}
Hence \eqref{5.30} and \eqref{5.33} apply to
the stationary KdV solutions
of Section 3 (e.g., \eqref{5.33} for $j=1$
coincides with \eqref{3.18})
which, in general, are quasi-periodic with
respect to $x$.  Moreover,
\eqref{5.30} and \eqref{5.33} also extend to
certain classes of
almost-periodic $V(x)$, see, e.g., \cite{8},
\cite{28}, \cite{31},
\cite{32}, Chs.~9, 11.

The periodic Dirichlet trace formula
\eqref{5.33} for $j=1$ has been
noticed by Hochstadt \cite{25} and later
on by Dubrovin \cite{10}.  The
general case $j=\bbN$ appeared in McKean
and van Moerbeke \cite{34} and
Flaschka \cite{13}.  More recent accounts
of \eqref{5.33} can be found in
\cite{8}, \cite{28}, \cite{31}, \cite{32},
Chs.~9, 11, \cite{40}.  The
Neumann case $\beta =0$ in \eqref{5.30} is
due to McKean and Trubowitz
\cite{35}.  The general case $\beta \in\bbR$
was first studied in
\cite{24}.  Additional references on the
subject of trace formulas and
their use in connection with the inverse
spectral problem can be found in
the papers cited in this paragraph and in
the ones listed at the
beginning of this section.

\appendix
\section{Hyperelliptic Curves of the
KdV-type and Theta Functions}
\renewcommand{\theequation}{A.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{thm}{0}

We briefly summarize our basic notation for
hyperelliptic KdV-type curves
(i.e., those branched at infinity) and their
theta functions as employed
in Sections~3 and 4.  For details on this
standard material we refer,
e.g., to \cite{11}, \cite{12}, \cite{29},
\cite{37}.

Consider the points $\{ E_n \}_{0 \leq n \leq g}
\subset \bbR$,
$E_0 < E_1 < \cdots < E_{2g}$, $g\in\bbN_0$ and
define the cut plane
$\Pi = \bbC \bs \bigcup_{j=0}^{g-1} [E_{2j},
E_{2j+1}] \cup [E_{2g},
\infty)$ with the holomorphic function
\begin{equation}
R_{2g+1} (.)^{1/2}: \left\{\begin{array}{l}
\Pi \to \bbC \\
z\to \big[\prod_{n=0}^{2g} (z-E_n)\big]^{1/2}
\end{array}\right.
\lb{a.3}
\end{equation}
on it.  $R_{2g+1} (.)^{1/2}$ is extended to
all of $\bbC$ by
\begin{equation}
R_{2g+1} (\lam)^{1/2} =
\lim\limits_{\eps \downarrow 0} R_{2g+1} (\lam +
i \eps)^{1/2}, \quad \lam \in\bbC \bs \Pi,
\lb{a.4}
\end{equation}
with the sign of the square root chosen
according to
\begin{equation}
R_{2g+1} (\lam)^{1/2} = \left\{ \begin{array}{ll}
(-1)^g i |R_{2g+1} (\lam)^{1/2} |,
& \lam \in (-\infty, E_0)\\
(-1)^{g+j} i |R_{2g+1} (\lam )^{1/2} |,
& \lam \in (E_{2j-1}, E_{2j}),
\quad 1\leq j \leq g\\
(-1)^{g+j} |R_{2g+1} (\lam)^{1/2}|,
& \lam \in (E_{2j}, E_{2j+1}), \quad
0 \leq j \leq g -1\\
|R_{2g+1} (\lam)^{1/2} |,
& \lam \in (E_{2g}, \infty)
\end{array}\right..
\lb{a.5}
\end{equation}
Next we define the set
\begin{equation}
M = \{ ( z, \sig R_{2g+1} (z)^{1/2} ) | z\in\bbC,
\; \sig \in \{-, +\}
\} \cup \{ P_\infty = (\infty, \infty) \}
\lb{a.6}
\end{equation}
and
\begin{equation}
B= \{(E_n, 0)\}_{0 \leq n \leq 2g }
\cup \{ P_\infty = (\infty,
\infty)\},
\lb{a.7}
\end{equation}
the set of branch points.  $M$ becomes a compact
 Riemann surface upon
introducing the charts $(U_{P_0}, \zeta_{P_0})$
defined as follows
\begin{align}
\begin{split}
P_0 & = (z_0, \sig_0 R_{2g+1} (z)^{1/2} )
 \mbox{ or } P_0 = P_\infty,\\
P& = (z, \sig R_{2g+1} (z)^{1/2} )
\in U_{P_0} \subset M, \; V_{P_0} =
\zeta (U_{P_0}) \subset \bbC.
\lb{a.8}
\end{split}
\end{align}

\medskip
\noindent\underline{{\boldmath $P_0 \in
M\bs B$:}}\\
$U_{P_0}  = \{ P\in M |\, |z-z_0| < C,\;
\sig R_{2g+1} (z)^{1/2}$
the branch obtained
by straight line analytic continuation
starting from $z_0 \}$,
$C = \min_n |z_0 - E_n|$,
\begin{equation}
\zeta_{P_0} : \left\{\begin{array}{l}
U_{P_0} \to V_{P_0}\\
P\to (z-z_0)
\end{array}\right.,\quad \zeta_{P_0}^{-1} :
\left\{\begin{array}{l}
V_{P_0} \to U_{P_0}\\
\zeta \to (z_0 + \zeta, \sig R_{2g+1} (z_0
+ \zeta)^{1/2})
\end{array}\right..
\lb{a.9}
\end{equation}

\medskip
\noindent\underline{{\boldmath $P_0 =
 (E_{n_0}, 0)$:}}
{\allw
\begin{align}
\begin{split}
& U_{P_0}  = \{ P\in M | \,
|z-E_{n_0} | < C_{n_0} \},\quad
C_{n_0}  = \left\{ \begin{array}{ll}
\min\limits_{n\neq n_0} |E_n - E_{n_0} |,
& g\in\bbN\\
\infty, & g=0
\end{array}\right.,\\
& V_{P_0}  = \{ \zeta \in\bbC |\,
|\zeta | < C_{n_0}^{1/2} \},\quad
\zeta_{P_0} : \left\{ \begin{array}{l}
U_{P_0} \to V_{P_0}\\
P\to \sig (z-E_{n_0})^{1/2}
\end{array}\right.,\\
& (z-E_{n_0})^{1/2} = |(z-E_{n_0} )^{1/2}
 | e^{(i/2) \arg (z-E_{n_0})},
\;
\arg (z-E_{n_0}) \in \left\{\begin{array}{ll}
[0, 2\pi), & n_0 \mbox{ even}\\
(-\pi, \pi], & n_0 \mbox{ odd}
\end{array}\right.,\\
&\zeta_{P_0}^{-1} : \left\{\begin{array}{l}
V_{P_0} \to U_{P_0} \\
\zeta \to (E_{n_0} +\zeta^2, \;
\zeta [\prod_{n\neq  n_0} (E_{n_0} -
E_n
+ \zeta^2)]^{1/2}
\end{array}\right.,\\
& \big[\prod_{n\neq n_0} (E_{n_0} - E_n
+\zeta^2 )\big]^{1/2} = (-1)^g i^{-n_0}
\big| \big[ \prod_{n\neq n_0} (E_{n_0} -
 E_n ) \big]^{1/2} \big|
\big[ 1 + 2^{-1} \zeta^2 \sum_{n\neq n_0
} (E_{n_0} - E_n
)^{-1} + 0 (\zeta^4)\big].
\lb{a.10}
\end{split}
\end{align}
}

\medskip
\noindent \underline{{\boldmath
$P_0 = P_\infty$:}}
{\allw
\begin{align}
\begin{split}
& U_{P_0} = \{P \in M |\, |z| > C_\infty \},\;
C_\infty = \max\limits_{n} |E_n|,\;
V_{P_0} = \{ \zeta \in\bbC |\,
 |\zeta| < C_\infty^{-1/2}\},\\
& \zeta_{P_0} : \textstyle\left\{\begin{array}{l}
U_{P_0} \to V_{P_0}\\
P \to \sig (1/z^{1/2})\\
P_\infty \to 0
\end{array}\right., \quad
\begin{array}{l}
z^{1/2} = |z^{1/2}| e^{(i/2) \arg (z)},\\
0 \leq \arg (z) < 2\pi,
\end{array}\\
& \zeta_{P_0}^{-1} : \begin{cases}
V_{P_0} \to U_{P_0}\\
\zeta \to \big(\zeta^{-2}, \zeta^{-2g-1}
 \big[ \prod_n (1-\zeta^2
E_n)\big]^{1/2}\big)\\
0 \to P_\infty
\end{cases},\\
& \big[ \prod_n (1-\zeta^2 E_n)\big]^{1/2}
= 1-2^{-1} \zeta^2 \sum_n E_n
+ 0 (\zeta^4).
\lb{a.11}
\end{split}
\end{align}
}

Upper and lower sheets $\Pi_\pm \subset M$
with associated charts
$\zeta_\pm$ are defined by
\begin{equation}
\Pi_\pm = \{ (z, \pm R_{2g+1} (z)^{1/2} )
\in M | z\in \Pi\},\;
\zeta_\pm:
\begin{cases}
\Pi_\pm \to \Pi\\
P \to z
\end{cases}.
\lb{a.12}
\end{equation}
The compact Riemann surface (curve) resulting
from
\eqref{a.6}--\eqref{a.11} is denoted by $K_g$.
Topologically, $K_g$ is a
sphere with $g$ handles and hence has genus $g$.

Next, define the holomorphic sheet exchange
 map (involution)
\begin{equation}
*: \begin{cases}
K_g \to K_g\\
(z,\sig R_{2g+1} (z)^{1/2}) \to (z,\sig R_{2g+1}
 (z)^{1/2})^* = (z, -\sig
R_{2g+1} (z)^{1/2} )
\end{cases}
\lb{a.13}
\end{equation}
and the two meromorphic projection maps
\begin{equation}
\ti \pi:\begin{cases}
K_g \to \bbC \cup \{ \infty\} \\
(z, \sig R_{2g+1} (z)^{1/2} ) \to z\\
P_\infty \to \infty
\end{cases}, \quad
R_{2g+1}^{1/2} : \begin{cases}
K_g \to \bbC \cup \{ \infty\}\\
(z,\sig R_{2g+1} (z)^{1/2} ) \to \sig R_{2g+1}
 (z)^{1/2}\\
P_\infty \to \infty
\end{cases}.
\lb{a.14}
\end{equation}
$\ti\pi$ has a pole of order $2$ at $P_\infty$ and
 two simple zeros at
$(0, \pm R_{2g+1} (0)^{1/2})$ if $R_{2g+1}(0)
\neq 0$ or a double zero at
$(0,0)$ if $R_{2g+1} (0) =0$ (i.e.,
if $0\in \{E_n\}_{0 \leq n \leq 2g}$)
and $R_{2g+1}^{1/2}$ has a pole of order
$2g+1$ at $P_\infty$ and $2g+1$
simple zeros at $(E_n, 0)$,
$0 \leq n \leq 2g$.  Moreover,
\begin{equation}
\ti \pi (P^*) = \ti \pi (P), \;
R_{2g+1}^{1/2} (P^*) = - R_{2g+1}^{1/2}
(P), \quad P \in K_g.
\lb{a.16}
\end{equation}
Thus $K_g$ is a two-sheeted ramified
covering of the Riemann sphere
$\bbC_\infty (\cong \bbC \cup \{ \infty\} )$,
in particular, $K_g$ is
compact and hyperelliptic.

Using our local charts one infers that for
$g\in\bbN$, $d\ti\pi /
R_{2g+1}^{1/2}$ is a holomorphic differential
on $K_g$ with a zero of
order $2(g-1)$ at $P_\infty$ and hence
\begin{equation}
\eta_j =
\ti\pi^{j-1} d \ti\pi / R_{2g+1}^{1/2} (.),
\quad 1\leq j \leq g
\lb{a.17}
\end{equation}
form a basis for the space of holomorphic
differentials on $K_g$.

Next we introduce a canonical homology basis
$\{ a_j, b_j\}_{1\leq j \leq
g}$ for $K_g$ as follows.  The cycle $a_\ell$
starts near $E_{2\ell -1}$
on $\Pi_+$, surrounds $E_{2\ell}$
counterclockwise thereby changing to
$\Pi_-$, and returns to the starting point
encircling $E_{2\ell -1}$,
changing sheets again.  The cycle $b_\ell$
surrounds $E_0$, $E_{2\ell
-1}$ counterclockwise (once) on $\Pi_+$.
The cycles are chosen such
that their intersection matrix reads $a_j \circ b_k
= \del_{j,k}$,
$1\leq j,k \leq g$. Introducing the invertible
matrix $C$ in $\bbC^g$,
\begin{align}
\begin{split}
C & =(C_{j,k})_{1\leq j,k \leq g}, \; C_{j,k}
= \int_{a_k} \eta_j =
2\int_{E_{2k-1}}^{E_{2k}} z^{j-1} \, dz
/ R_{2g+1} (z)^{1/2} \in i\bbR,\\
\underline{c} (k) & = (c_1(k), \ldots,
c_g(k)), \quad c_j (k) =
C_{j,k}^{-1},
\lb{a.19}
\end{split}
\end{align}
the normalized differentials $\ome_j$,
$1\leq j \leq g$,
\begin{equation}
\ome_j = \sum_{\ell=1}^g c_j (\ell) \eta_\ell,
\; \int_{a_k} \ome_j =
\del_{j,k}, \quad 1\leq j,k \leq g
\lb{a.20}
\end{equation}
form a canonical basis for the space of
holomorphic differentials on
$K_g$.  The matrix $\tau$ in $\bbC^g$ of
$b$-periods,
\begin{equation}
\tau = (\tau_{j,k})_{1\leq j,k\leq g},
\; \tau_{j,k} = \int_{b_k}
\ome_j, \quad 1\leq j,k\leq g
\lb{a.21}
\end{equation}
then satisfies
\begin{equation}
 \tau_{j,k} = \tau_{k,j},
\quad 1\leq j,k \leq g,\;
 \tau = iT, \quad T > 0.
\lb{a.22}
\end{equation}
In the chart $(U_{P_\infty}, \zeta_{P_\infty}
 = \zeta)$ induced by $1/
\ti \pi^{1/2}$ near $P_\infty$ one infers
\begin{align}
\begin{split}
\uome & = -2 \big\{ \sum_{j=1}^g \uc (j)
\zeta^{2(g-j)} / \big[ \prod_n
(1-\zeta^2 E_n) \big]^{1/2} \big\}
\, d\zeta\\
& = -2 \big\{ \uc (g) +
\big[ \textstyle\frac12 \uc (g)
\sum_{n=0}^{2g} E_n +
 \uc
(g-1) \big] \zeta^2 + 0 (\zeta^4) \big\}
 \, d\zeta.
\lb{a.24}
\end{split}
\end{align}
Associated with the homology basis
$\{a_j, b_j\}_{1\leq j \leq g}$ we
also recall the canonical dissection
of $K_g$ along its cycles yielding
the simply connected interior $\hat K_g$
of the fundamental polygon $\pa
\hat K_g$ given by $\pa \hat K_g =
a_1 b_1 a_1^{-1} b_1^{-1}
a_2 b_2a_2^{-1} b_2^{-1}
 \cdots a_g^{-1} b_g^{-1}$. The Riemann
theta function associated with $K_g$ is
defined by
\begin{equation}
\theta (\uz) = \sum_{\un \in\bbZ^g}
 \exp [2\pi i (\un,\uz) + \pi i (\un,
\tau \un)], \quad \uz =(z_1,
 \ldots, z_g) \in\bbC^g,
\lb{a.26}
\end{equation}
where $(\underline{u}, \uv) =
\sum_{j=1}^g  \overline{u_j} v_j$ denotes the
scalar product in $\bbC^g$.  It has
the fundamental properties
\begin{align}
\begin{split}
& \theta (z_1, \ldots, z_{j-1}, - z_j, z_{j+1},
\ldots, z_g) = \theta
(\uz),\\
& \theta (\uz + \um + \tau \un) =
\theta (\uz) \exp[-2\pi i (\un,\uz) -
\pi i (\un, \tau \un)],
\quad \um, \un \in \bbZ^g.
\lb{a.27}
\end{split}
\end{align}
A divisor $\calD$ on $K_g$ is a map $\calD:
K_g \to \bbZ$, where $\calD
(P) \neq 0$ for only finitely-many $P\in K_g$.
The set of all divisors
will be denoted by $\ddiv (K_g)$.
With $L_g$ we denote the period
lattice
\begin{equation}
L_g = \{ \uz \in\bbC^g | \uz = \um +
\tau \un, \; \um, \un \in\bbZ^g\}
\lb{a.28}
\end{equation}
and the Jacobi variety $J(K_g)$ is
defined by
\begin{equation}
J(K_g) = \bbC^g / L_g.
\end{equation}
The Abel maps $\ua_{P_0} (.)$, respectively
$\ual_{P_0} (.)$ are defined
by
\begin{equation}
\ua_{P_0} : \begin{cases} \scriptstyle
K_g \to J(K_g) \\ \scriptstyle
P \to \ua_{P_0} (P) =
\int\limits_{P_0}^P \uome \! \mod (L_g)
\end{cases}, \quad
\ual_{P_0} : \begin{cases} \scriptstyle
\ddiv (K_g) \to J(K_g) \\ \scriptstyle
\calD \to \ual_{P_0} (\calD) =
\sum\limits_{P \in K_g} \calD (P)
\ua_{P_0} (P) \end{cases},
\lb{a.29}
\end{equation}
with $P_0 \in K_g$ a fixed base point.
(In the main text we agree to fix
$P_0 = (E_0, 0)$ for convenience.)

Next, let $\calM (K_g)$ and $\calM^1 (K_g)$
denote the set of meromorphic
functions ($0$-forms) and meromorphic
differentials
 ($1$-forms) on $K_g$.
The residue of a meromorphic differential
$\nu \in\calM^1 (K_g)$ at a
point $Q_0 \in K_g$ is defined by
$\operatornamewithlimits{res}_{Q_0} (\nu)
= (2 \pi i)^{-1}
\int_{\gam_{Q_0}} \nu$, where $\gam_{Q_0}$ is
 a counterclockwise
oriented smooth simple closed
contour encircling $Q_0$ but no other pole of
$\nu$.  Holomorphic
differentials are also called (Abelian)
differentials of the first kind
(dfk); (Abelian) differentials of the second
kind (dsk) $\ome^\btwo \in
\calM^1 (K_g)$ are characterized by the property
that all their residues
vanish.  They are normalized, e.g., by demanding
that all their
$a$-periods vanish, i.e.,
\begin{equation}
\int_{a_j} \ome^\btwo =0,\quad 1\leq j \leq g.
\lb{a.33}
\end{equation}
If $\ome_{P_1, n}^\btwo$ is a dsk on $K_g$ whose
only pole is $P_1 \in
\hat K_g$ with principal part $\zeta^{-n-2}
\, d\zeta$,
$n\in\bbN_0$ near
$P_1$ and $\ome_j= (\sum_{m=0}^\infty d_{j,m}
 (P_1) \zeta^m) \, d\zeta$
near
$P_1$, then
\begin{equation}
\int_{b_j} \ome_{P_{1,n}}^\btwo = [2\pi i
/ (n+1) ] d_{j,n} (P_1).
\lb{a.34}
\end{equation}
A basis for dsk's on $K_g$, holomorphic
on $K_g \bs \{ P_\infty\}$, is
provided by
\begin{equation}
\ome_n^\btwo = \ti \pi^{g+1+n} d\ti \pi
/ R_{2g+1}^{1/2} (.),\quad
n\in\bbN_0.
\lb{a.35}
\end{equation}
Any meromorphic differential $\ome^{\bth}$
on $K_g$ not of the first or
second kind is defined to be of the third
kind $(\dtk)$.  A $\dtk\;
\ome^\bth
\in \calM^1 (K_g)$ is usually normalized by
the vanishing of its
$a$-periods, i.e.,
\begin{equation}
\int_{a_j} \ome^\bth =0,\quad 1\leq j\leq g.
\lb{a.36}
\end{equation}
A normal $\dtk \; \ome_{P_1, P_2}^\bth$
associated with two points $P_1,
P_2
\in\hat K_g$, $P_1 \neq P_2$ by definition has
simple poles at $P_1$ and
$P_2$ with residues $+1$ at $P_1$ and $-1$ at
$P_2$ and vanishing
$a$-periods.  If $\ome_{P,Q}^\bth$ is a normal
dtk associated with $P$,
$Q\in \hat K_g$, holomorphic on
$K_g \bs \{ P, Q\}$, then
\begin{equation}
\int_{b_j} \ome_{P,Q}^\bth =
2\pi i \int_Q^P \ome_j
, \quad 1 \leq j \leq
g,
\lb{a.37}
\end{equation}
where the path from $Q$ to $P$ lies in
$\hat K_g$ (i.e., does not touch
any of the cycles $a_j$, $b_j$).

We shall always assume (without loss of
generality) that all poles of
dsk's and dtk's on $K_g$ lie on $\hat K_g$
(i.e., not on $\pa \hat K_g$).

For $f\in \calM (K_g) \bs \{0\}$,
$\ome\in \calM^1 (K_g) \bs \{0\}$ the
divisors of $f$ and $\ome$ are denoted by
$(f)$ and $(\ome)$,
respectively.  Two divisors $\calD$,
$\calE\in \ddiv (K_g)$ are called
equivalent, denoted by $\calD \sim \calE$,
if and only if $\calD - \calE
=(f)$ for some $f\in \calM (K_g) \bs \{0\}$.
The divisor class $[\calD]$
of $\calD$ is then given by $[\calD] =
\{\calE \in \ddiv (K_g)| \calE
\sim
\calD \}$.  We recall that
\begin{equation}
\deg ((f)) =0, \; \deg ((\ome)) = 2(g-1),
\quad f\in \calM (K_g) \bs\{
0\},\; \ome\in\calM^1 (K_g) \bs \{0\},
\lb{a.38}
\end{equation}
where the degree $\deg (\calD)$ of
$\calD$ is given by $\deg (\calD) =
\sum_{P\in K_g} \calD(P)$.  One calls
 $(f)$ (respectively $(\ome)$) a
principal (respectively canonical) divisor.

Introducing the complex linear spaces
\begin{align}
\calL (\calD) & = \{ f\in\calM (K_g) | f=0
 \mbox{ or } (f) \geq \calD \},
\; r(\calD) = \dim_\bbC \calL (\calD),
\lb{a.39}\\
\calL^1 (\calD)&= \{ \ome \in
\calM^1 (K_g) | \ome = 0 \mbox{ or } (\ome)
\geq \calD \},\; i (\calD) =
 \dim_\bbC \calL^1 (\calD)
\lb{a.40}
\end{align}
($i(\calD)$ the index of speciality of
$\calD$), one infers that $\deg
(\calD)$, $r(\calD)$, and $i(\calD)$ only
depend on the divisor class
$[\calD]$ of $\calD$.  Moreover, we recall
the following fundamental
facts.

\begin{thm} \lb{ta.1}
Let $\calD \in \ddiv (K_g)$, $\ome \in
\calM^1 (K_g)\bs \{0\}$. Then\\
(i).
\begin{equation}
i(\calD) = r(\calD- (\ome)), \quad g\in\bbN_0.
\lb{a.41}
\end{equation}
(ii). (Riemann-Roch theorem).
\begin{equation}
r(-\calD) = \deg (\calD) + i(\calD) -g+1,
\quad g\in\bbN_0.
\lb{a.42}
\end{equation}
(iii).\quad (Abel's theorem).  $\calD \in
\ddiv (K_g)$, $g\in \bbN$ is
principal if and only if
\begin{equation}
\deg (\calD ) = 0 \mbox{ and } \ual_{P_0}
 (\calD) =\uzero.
\lb{a.43}
\end{equation}
(iv).\quad (Jacobi's inversion theorem).
Assume $g\in\bbN$, then
$\ual_{P_0} : \ddiv (K_g) \to J(K_g)$ is
surjective.
\end{thm}

For notational convenience we agree to
abbreviate
\begin{equation}
\calD_Q : \begin{cases}
K_g \to \{0,1\}\\
P \to \begin{cases}
1, & P= Q\\ 0, & P\neq Q
\end{cases}
\end{cases}, \quad
\calD_{\uq}: \begin{cases}
K_g \to \{0,1,\ldots, g\}\\
P \to \begin{cases}
m & \mbox{ if } P \mbox{ occurs } m\mbox{-times in }
 \{Q_1, \ldots, Q_g
\}\\ 0 & \mbox{ if } P \not\in
\{Q_1, \ldots, Q_g\}
\end{cases}
\end{cases}.
\lb{a.44}
\end{equation}
for $\uq=(Q_1,\ldots, Q_g) \in\sig^g K_g$
 ($\sig^n K_g$ then $n$-th
symmetric power of $K_g$). Moreover,
$\sig^n K_g$  can be identified
with the set of positive
divisors $0< \calD \in\ddiv (K_g)$ of
degree $n$.

\begin{lem} \lb{la.2}
Let $\calD_{\uq} \in \sig^g K_g$, $\uq =
 (Q_1, \ldots, Q_g)$.  Then
$1 \leq i (\calD_Q) = s(\leq g/ 2)$
if and only if there are $s$ pairs of the
type $(P, P^*) \in \{ Q_1,
\ldots, Q_g\}$ (this includes, of course,
branch points for which $P=
P^*$).
\end{lem}

We emphasize that most results in this
appendix immediately extend to the
case where $\{E_n\}_{0 \leq n \leq 2g}
\subset \bbC$.  (In this case
$\tau$ is no longer purely imaginary as
stated in \eqref{a.22} but has a
positive definite imaginary part.)

\section{An Explicit Illustration of the
Riemann-Roch Theorem}
\renewcommand{\theequation}{B.\arabic{equation}}
\setcounter{equation}{0}
\setcounter{thm}{0}

Finally, we give a brief illustration of
the Riemann-Roch theorem in
connection with KdV-type hyperelliptic curves,
i.e., hyperelliptic curves
branched at infinity, and explicitly determine
a basis for the vector
space $\calL (-n\calD_{P_\infty} -
 \calD_{\humu (x_0)} )$, $n\in\bbN_0$.

We freely use the notation introduced in
Appendix~A and refer, in
particular, to the definition \eqref{a.39}
of $\calL (\calD)$ and the
Riemann-Roch theorem stated in
Theorem~\ref{ta.1} (ii).  In addition, we
use the short-hand notation
\begin{equation}
n \calD_{P_\infty} + \calD_{\humu (x_0)} =
 \sum_{m=1}^n \calD_{P_\infty}
+ \sum_{j=1}^g \calD_{\hat \mu_j (x_0)}, \quad
n\in\bbN_0,\; \humu (x_0) = (\hat \mu_1 (x_0),
\ldots, \hat \mu_g (x_0))
\lb{b.1}
\end{equation}
and recall that
\begin{equation}
\calL (-n\calD_{P_\infty} -\calD_{\humu(x_0)}) =
 \{ f\in\calM (K_g)| f=0
\mbox{ or } (f) + n\calD_{P_\infty} +
 \calD_{\humu (x_0)} \geq 0\}, \quad
n \in\bbN_0.
\lb{b.2}
\end{equation}
With $\phi (P, x)$, $\psi (P,x,x_0)$
defined as in \eqref{3.14},
\eqref{3.17} we obtain the following

\begin{thm} \lb{tb.1}
Assume $\calD_{\humu(x_0)}$ to be nonspecial
(i.e., $i( \calD_{\humu
(x_0) } ) =0$) and of degree $g\in \bbN$.
For $n\in\bbN_0$, a basis for
the vector space $\calL (-n \calD_{P_\infty}
- \calD_{\humu (x_0)})$ is
given by
\begin{equation}
\begin{cases}
\{1\}, & n=0\\
\{ \ti \pi^j\}_{0 \leq j \leq (n-1) /2}
\cup \{ \ti\pi^j \phi (.,x_0)
\}_{0\leq j \leq (n-1)/2}, & n \mbox{ odd}\\
\{ \ti \pi^j \}_{0 \leq j \leq n/2}
\cup \{ \ti \pi^j \phi (.,x_0) \}_{0
\leq j \leq (n-2)/2}, & n \mbox{ even}
\end{cases},
\lb{b.3}
\end{equation}
or equivalently,
\begin{equation}
\calL (-n \calD_{P_\infty} - \calD_{\humu(x_0)} )
= \operatorname{span}
\big\{
\dfrac{\pa^j}{\pa x^j} \psi (., x, x_0)
 \big|_{x=x_0} \big\}_{0 \leq j
\leq n}.
\lb{b.4}
\end{equation}
\end{thm}

\begin{proof}
The elements in \eqref{b.3} easily seen to
be linearly independent and
belonging to $\calL (-n\calD_{P_\infty} -
 \calD_{\humu (x_0)} )$.  It
remains to be shown that they are maximal.
{}From $0= i( \calD_{\humu (x_0)})
= i (\calD_{n P_\infty} + \calD_{\humu (x_0)})$
and the Riemann-Roch
theorem \eqref{a.42} one obtains
$r(-n\calD_{P_\infty} - \calD_{\humu(x_0)})
= n+1$
proving \eqref{b.3}.  In order to prove
\eqref{b.4}, one repeatedly uses
the \schro equation \eqref{3.20} to prove
inductively that
\begin{align}
\begin{split}
& \dfrac{\pa^{2m+2}}{\pa x^{2m+2}} \psi
(P,x,x_0) = (-\ti \pi)^{m+1} +
R_{2m+1} (P,x),\\
& \dfrac{\pa^{2m+1}}{\pa x^{2m+1}}
\psi (P,x,x_0) = (- \ti\pi)^m
\dfrac{\pa}{\pa x} \psi (P,x,x_0) +
R_{2m} (P,x),
\lb{b.6}
\end{split}
\end{align}
where $R_n (.,x_0) \in \calL (-n \calD_{P_\infty}
- \calD_{\humu(x_0)})$.
\end{proof}

\section*{Acknowledgments}

F.~G.\ would like to thank the organizers for
their kind invitation to a
most stimulating conference.



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\end{document}

